Datasets:

zai-org
/

RPC-Bench

+# Neurocoder: Learning General-Purpose Computation Using Stored Neural Programs
+Anonymous Author(s)
+Affiliation
+Address
+email
+# Abstract
+Artificial Neural Networks are functionally equivalent to special-purpose computers. Their inter-neuronal connection weights represent the learnt Neural Program that instructs the networks on how to compute the data. However, without storing Neural Programs, they are restricted to only one, overwriting learnt programs when trained on new data. Here we design Neurocoder, a new class of generalpurpose neural networks in which the neural network “codes” itself in a dataresponsive way by composing relevant programs from a set of shareable, modular programs stored in external memory. For the first time, a Neural Program is efficiently treated as a datum in memory. Integrating Neurocoder into current neural architectures, we demonstrate new capacity to learn modular programs, reuse simple programs to build complex ones, handle pattern shifts and remember old programs as new ones are learnt, and show substantial performance improvement in solving object recognition, playing video games and continual learning tasks.
+# 14 1 Introduction
+15 From its inception in 1943 until recently, the fundamental architectures of Artificial Neural Net
+16 works remained largely unchanged - a program is executed by passing data through a network of
+17 artificial neurons whose inter-neuronal connection weights are learnt through training with data.
+18 These inter-neuronal connection weights, or Neural Programs, correspond to a program in modern
+19 computers [32]. Memory Augmented Neural Networks (MANN) are an innovative solution allow
+20 ing networks to access external memory for manipulating data [11, 12]. But they were still unable
+21 to store Neural Programs in such external memory, and this severely limits machine learning. Stor
+22 ing inter-neuronal connection weights only in their network does not permit modular separation of
+23 Neural programs and is analogous to a computer with one fixed program. Recent works introduce
+24 conditional computation via adjusting or activating parts of a network in an input-dependent manner
+25 [39, 33, 4, 13, 28], but networks remain monolithic. Current networks forget when retrained, old
+26 inter-neuronal connection weights are merged with new ones or erased.
+27 The brain is modular, not a monolithic system [8, 6]. Neuroscience research indicates that the brain is
+28 divided into functional modules [19, 7, 9]. If the neural program for each module is kept in separate
+29 networks, networks proliferate. Modular neural networks, another form of conditional computation,
+30 combine the output of multiple expert networks, but as the experts grow, the networks grow drasti
+31 cally [20, 14, 35, 29]. This requires huge computational storage and introduces redundancy as these
+32 experts do not share common basic programs.
+33 A pathway out of this bind is to keep such basic programs in memory and combine them as required.
+34 This brings neural networks towards modern general-purpose computers that use the stored-program
+35 principle [37, 40] to efficiently access reusable programs in external memory. Here we show how
+36 Neurocoder, a new neural framework, introduces a new class of general-purpose conditional compu
+37 tation machines in which a neural network can be “coded” in an input-dependent manner. Efficient
+38 decomposition of Neural Programs creates shareable modular components that can reconstruct the
+39 whole program space. These components change their “shapes” based on training and are stored
+40 in an external Program Memory. Then, in a data-responsive way, a Program Controller retrieves
+41 relevant components to build the Neural Program. This is analogous to shape-shifting Lego bricks
+42 that can be reused to build unlimited shapes and structures (See Appendix Fig. 4).
+43 Using adaptive modular components vastly increases the learning capacity of the neural network
+44 by allowing re-utilisation of parameters, effectively curbing network growth as programs increase.
+45 More importantly, unlike pre-defined sub-networks or modules [20, 1] that combine at activation
+46 level, the construction of our modular components is dynamic and performed on the weight space.
+47 The Neural Program construction is learnt through training via traditional backpropagation [30] as
+48 the architecture is end-to-end differentiable.
+![](images/e389c763e9b4533f0b062c87f24ebd5da2780f7e18d1c6818a55a57ce1880a0c.jpg)
+Figure 1: Neurocoder (a) The Main Network uses a working program to compute the output for the input. Here only the final layer of the Main Network is adaptively loaded with the working program $( I )$ . Other layers use traditional Neural Programs as connection weights (fixed-after-training). (b) The Program Controller’s composition network controls access to the Program Memory, emitting queries and interpolating gate control signals in response to the input (2). It then performs recurrent multi-head program attention to the Program Status (3), triggering attention weights to the Singular Programs (4). The attended Singular Programs form an active program using low-rank approximation (5). Residual program produced by the Program Controller’s integration network (6) plus the active program derives the working program. (c) The Program Memory stores the representations (singular programs) required to reconstruct the active program to be used by the Program Controller. Access is controlled through the Program Status including keys $( k )$ , and slot usage $( m )$ that are updated during the training and computation (7).
+# 49 2 Methods
+# 2.1 System overview
+51 A Neurocoder is a neural network (Main Network) coupled to an external Program Memory through
+52 a Program Controller. The working program of the Main Network processes the input data to pro
+53 duce the output. This working program is “coded” by the Program Controller by creating an input
+54 dependent active program from the Program Memory (Fig. 1). The following gives a high-level
+55 description of the Neurocoder framework and then the details.
+# 56 Neurocoder stores Singular Value Decomposition of Neural Programs in Program Memory
+57 The Neural Program needs to be stored efficiently in Program Memory. This is challenging as there
+58 may be millions of inter-neuronal connection weights, thus storing them directly ([22]) is grossly
+59 inefficient. Instead, the Neurocoder forms the basis of a subspace spanned by Neural Programs and
+60 stores the singular values and vectors of this subspace in memory slots of the Program Memory
+61 (hereafter referred to as singular programs). Based on the input, relevant singular programs are
+62 retrieved, a new program is reconstructed and then loaded in the Main Network to process the input.
+63 This representational choice significantly reduces the number of stored elements and allows each
+64 singular program to effectively represent a unitary function of the active program.
+65 The active program matrix $\mathbf { P }$ can be composed by standard low-rank approximation as
+$$
+\mathbf { P } = \mathbf { U S V } ^ { \mathbf { T } } = \sum _ { n } ^ { r _ { m } } \sigma _ { n } u _ { n } v _ { n } ^ { \top }
+$$
+66 where $\mathbf { U }$ and $\mathbf { V }$ are matrices of the left and right singular vectors, and S the matrix of singular values.
+67 $r _ { m }$ is the total number of components we want to retrieve. $\left\{ \sigma _ { n } \right\} _ { n = 1 } ^ { r _ { m } }$ is the attended singular values,
+68 {un} mn= and $\{ v _ { n } \} _ { n = 1 } ^ { r _ { m } }$ the attended singular vectors of S, U, and $\mathbf { V }$ , respectively. The Program
+69 Memory is crafted as three singular program memories $\{ { \bf M } _ { U } , { \bf M } _ { V } , { \bf M } _ { S } \}$ –each of their memory
+70 slot stores a singular component or singular program. The process “codes” the active program
+71 using singular programs from the program memories. The coding is conditioned on input $x _ { t }$ , yet
+72 we drop index $t$ for notation simplification and leave the details on the computation of $\sigma _ { n } , u _ { n } , v _ { n }$ in
+73 Sec. 2.2.
+74 The Program Memory also maintains the status for each singular program in terms of access and
+75 usage. To access a singular program, program keys $( k )$ are used. These keys are low-dimensional
+76 vectors that represent the singular program function and computed by a neural network that ef
+77 fectively compresses the singular program. The program usage $( m )$ measures memory utilisation,
+78 recording how much a memory slot is used in constructing a program. The components of the
+79 Program Memory are summarised in Fig. 1 (c).
+80 Recurrent multi-head program attention mechanisms for program storage and retrieval
+81 Neural networks use the concept of differentiable attention to access memory [11, 2]. This de
+82 fines a weighting distribution over the memory slots essentially weighting the degree to which each
+83 memory slot participates in a read or write operation. This is unlike conventional computers that use
+84 a unique address to access a single memory slot.
+85 Here we use two kinds of attention. First is content-based attention [11, 12] to ensure that the singu
+86 lar program is selected based on its functionality and the data input. This is achieved by producing
+87 a query vector based on the input and comparing it to the program keys $( k )$ using cosine similarity.
+88 Higher cosine similarity scores indicate higher attention weights to the singular programs associated
+89 with those program keys. Second, to encourage better memory utilisation, higher attention weights
+90 are assigned to slots with lower program usage $( m )$ through usage-based attention [12, 31]. The
+91 attention weights from the two schemas are then combined using interpolating gates to compose the
+92 final attention weights to the Program Memory.
+93 We adapt multi-head attention [11, 38] that applies multiple attentions in parallel to retrieve $H$ singu
+94 lar components. Besides, we introduce a recurrent attention mechanism, in which multi-head access
+95 is performed recurrently in $J$ steps. The $j$ -th set of $H$ retrieved components is conditioned on the
+96 previous ones. This recurrent, multi-head attention allows the composition network to incrementally
+97 search for optimal components for building relevant active programs.
+# 98 Neurocoder learns to “code” a relevant working program via training
+99 The structure of the Program Memory and the role of the Program Controller facilitates the au
+100 tomatic construction of working programs via training. The Program Controller controls memory
+101 access through its composition network that creates the attention weight defining how to weight the
+102 singular programs in the memories. A weighted summation of the singular programs results in the
+103 attended singular program. Applying the recurrent multi-head attention described earlier, multiple
+104 attended singular programs are retrieved to construct an active program (Eq. 1). Then the Program
+105 Controller generates a residual program using its integration network, adding to the active program
+106 to produce the working program of the Main Network. This addition enables creation of flexible
+107 higher-rank working programs, which compensates for the low-rank coding process. The structure
+108 of the Program Controller is illustrated in Fig. 1 (b).
+109 The singular programs are trained to represent unitary functions necessary for any computation
+110 whilst the composition and integration networks are trained to compose the relevant programs for
+111 the considering task. As such, beside minimising the task loss, we enforce orthogonality of stored
+112 singular vectors by minimising $\mathcal { L } _ { o } = \mathbf { M } _ { U } \mathbf { M } _ { U } ^ { \top } - \mathbf { \bar { I } } + \mathbf { M } _ { V } \mathbf { M } _ { V } ^ { \top } - \mathbf { I }$ . The parameters of the networks,
+113 and the stored singular programs are adjusted using gradient training via minimising the total loss
+$$
+\mathscr { L } = \mathcal { L } _ { t a s k } + a \mathcal { L } _ { o }
+$$
+where $\mathcal { L } _ { t a s k }$ represents the supervised task loss and $\mathcal { L } _ { o }$ represents the orthogonal loss weighted by a hyper-parameter $a$ to enforce orthogonality of the singular vectors.
+# 2.2 Attention mechanisms for Program Memory
+117 Here we ddenoted as $w _ { i n } ^ { u , v , \sigma }$ progra)–the a attention mechanintion weight to the $i$ ms used in this paper. Given -th slot of the singular progra $w _ { i n } ^ { u }$ , e $w _ { i n } ^ { v }$ , i $w _ { i n } ^ { \sigma }$ $\mathbf { M } _ { U }$ o, $\mathbf { M } _ { V }$
+119 , we retrieve the $n$ -th singular vector as follows,
+$$
+u _ { n } = \sum _ { i = 1 } ^ { P _ { u } } w _ { i n } ^ { u } \mathbf { M } _ { U } \left( i \right)
+$$
+$$
+v _ { n } = \sum _ { i = 1 } ^ { P _ { v } } w _ { i n } ^ { v } \mathbf { M } _ { V } \left( i \right)
+$$
+120 For the singular values, we need to enforce $\sigma _ { 1 } > \sigma _ { 2 } > . . . > \sigma _ { r _ { m } } > 0$ , thus we retrieve using
+$$
+\sigma _ { n } = \left\{ \begin{array} { l l } { \mathrm { s o f t p l u s } \left( \sum _ { i = 1 } ^ { P _ { s } } w _ { i n } ^ { \sigma } \mathbf { M } _ { S } \left( i \right) \right) } & { n = r _ { m } } \\ { \sigma _ { n + 1 } + \mathrm { s o f t p l u s } \left( \sum _ { i = 1 } ^ { P _ { s } } w _ { i n } ^ { \sigma } \mathbf { M } _ { S } \left( i \right) \right) } & { n < r _ { m } } \end{array} \right.
+$$
+121 Here, $P _ { u } , P _ { v }$ and $P _ { s }$ are the number of memory slots of $\mathbf { M } _ { U }$ , $\mathbf { M } _ { V }$ and $\mathbf { M } _ { S }$ , respectively. In this
+122 paper, we set $P = P _ { u } = P _ { v } = P _ { s }$ as the number of memory slots of the Program Memory. We note
+123 that thead ese notations are speci, and an attention step ed for some data input , hence the full notatio $x _ { t }$ and the hould be $n$ later maps to an attention To simplify notations, we
+$h$ $j$ $w _ { t i j h } ^ { u , v , \sigma }$
+125 will drop $u , v , \sigma$ from now and describe the computation of a representative $w _ { t i j h }$ for any of the
+126 three program memories in the following parts.
+# 127 Recurrent Access to the Program Memory via the composition network
+128 To perform program attention, the Program Controller employs a composition network (denoted
+129 as $f _ { \theta } )$ ), which takes the current input $x _ { t }$ and produce program composition control signals $( \pmb { \xi } _ { t } ^ { p } )$ .
+130 If $f _ { \theta }$ performs all attentions concurrently via multi-head attention as in [11, 38], it may lead to
+131 program collapse [22]. To have a better control of the component formation and alleviate program
+132 collapse, we propose to recurrently attend to the program memory. To this end, we implement $f _ { \theta }$ as
+133 $\pmb { \xi } _ { t } ^ { p } = \left\{ \pmb { \xi } _ { t j } ^ { p } \right\} _ { j = 1 } ^ { J }$ al network (LST. At access step $j$ M [16]) and let it access the program memory , the recurrent network updates its hidden sta $J$ times, resultin and generates $\xi _ { t j } ^ { p }$
+135 using recurrent dynamics as
+$$
+{ \pmb { \xi } } _ { t j } ^ { p } , h _ { j } = f _ { \theta } \left( x _ { t } , h _ { j - 1 } \right)
+$$
+136 where $h _ { 0 }$ is initialized as zeros and $\xi _ { t j } ^ { p }$ is the program composition control signal at step $j$ that
+137 depends on both on the input data $x _ { t }$ and the the previous state $h _ { j - 1 }$ . Particularly, the control signal
+138 contains the queries and the interpolation gates for each head to compute the program attention
+139 weight: $\pmb { \xi } _ { t j } ^ { p } = \{ q _ { t j h } , g _ { t i j h } \} _ { h = 1 } ^ { H }$ . Here, at each attention step, we perform multi-head attention with
+140 $H$ as the number of attention heads and thus, each $\xi _ { t j } ^ { p }$ consists of $H$ pairs of queries and gates.
+141 Hence, the total number of retrieved components $\boldsymbol { r } _ { m } = \boldsymbol { \bar { J } } \times \boldsymbol { H }$ and the index $n = j \times H + h$ .
+# 142 Attending to Programs by “Name”
+143 Inspired by the content-based attention mechanism for data memory [11], we use the query to look
+144 for the singular programs. In computer programming, to find the appropriate program for some
+145 computation, we often refer to the program description or at least the name of the program. Here, we
+146 create the “name” for our neural programs by compressing the program content to a low-dimensional
+147 key vector. As such, we employ a neural network $( f _ { \varphi } )$ to compute the program memory keys as
+$$
+k _ { i } = f _ { \varphi } \left( \mathbf { M } \left( i \right) \right)
+$$
+148 where $\boldsymbol { k } _ { i } \in \mathbb { R } ^ { K }$ and $i$ is the row index of the program memory. Here, $f _ { \varphi }$ learns to compress each
+149 memory slot into a $K$ -dimensional vector. As the singular programs evolve, their keys get updated.
+150 In this paper, we update the program keys after each learning iteration during training.
+151 Finally the content-based program memory attention $c _ { t i j h }$ is computed using cosine distance be
+152 tween the program keys $k _ { i }$ and the queries $q _ { t j h }$ as
+$$
+c _ { t i j h } = \mathrm { s o f t m a x } ^ { ( i ) } \left( \frac { q _ { t j h } \cdot k _ { i } } { \left| \left| q _ { t j h } \right| \right| \cdot \left| \left| k _ { i } \right| \right| } \right)
+$$
+# 153 Making Every Program Count
+154 Similarly to [12, 31], in addition to the content-based attention, we employ a least-used reading
+155 strategy to encourage the Program Controller to assign different singular programs to different com
+156 ponents. In particular, we calculate the memory usage for each program slot across attentions as
+157
+$$
+m _ { t i j h } = \operatorname* { m a x } _ { \tilde { j } \leq j } \left( w _ { t i \tilde { j } h } \right)
+$$
+158 Since we want to consider only $l _ { I }$ amongst $P$ memory slots that have smallest usages, let $\hat { m } _ { t j h } ^ { l _ { I } }$
+159 denote the value of the $l _ { I }$ -th smallest usage, then the least-used attention is computed as
+$$
+l _ { t i j h } = \left\{ \begin{array} { l l } { \underset { i } { \operatorname* { m a x } } \left( m _ { t i j h } \right) - m _ { t i j h } } & { ; m _ { t i j h } \leq \hat { m } _ { t j h } ^ { l _ { I } } } \\ { 0 } & { ; m _ { t i j h } > \hat { m } _ { t j h } ^ { l _ { I } } } \end{array} \right.
+$$
+160 The final program memory attention is computed as
+$$
+w _ { t i j h } = \mathrm { s i g m o i d } \left( g _ { t i j h } \right) c _ { t i j h } + \left( 1 - \mathrm { s i g m o i d } \left( g _ { t i j h } \right) \right) l _ { t i j h }
+$$
+161 Since the usage record are computed along the memory accesses, the multi-step Neurocoder utilises
+162 this attention mechanism better than the single-step Neurocoder, creating different attention styles
+163 (see Sec. 3.2). The composition the active program $\mathbf { P } _ { t }$ is illustrated in Appendix’s Fig. 5.
+# 2.3 Program Integration via the integration network
+165 Since the working program $\mathbf { P } _ { t }$ only contains top $r _ { m }$ principal components, it is low-rank and may
+166 be not flexible enough for sophisticated computation. We propose to enhance $\mathbf { P } _ { t }$ with a residual
+167 program $\mathbf { R } -$ a traditional connection weight trained as the integration network’s parameters, which
+168 is constant after training w.r.t $t$ . The residual program represents the sum of the remaining less
+169 important components. To this end, we suppress $\mathbf { R }$ with a multiplier that is smaller than $\sigma _ { t r _ { m } } .$ – the
+170 smallest singular value of the main components - resulting in the integration formula
+$$
+W _ { t } = \mathbf { P } _ { t } + w _ { t } ^ { r } \sigma _ { t r _ { m } } \mathbf { R }
+$$
+171 where $w _ { t } ^ { r } = \mathrm { s i g m o i d } \left( f _ { \phi } \left( x _ { t } \right) \right)$ is an adaptive gating value that controls the contribution of the
+172 residual program. $f _ { \phi }$ is the integration network in the Program Controller and hence, in our imple
+173 mentation, the integration control signal sent by the Program Controller is $\lambda _ { t } ^ { p } = \{ w _ { t } ^ { r } , \sigma _ { t r _ { m } } \}$ . We
+174 note that in our experiments, the program integration can be disabled ( $W _ { t }$ is directly set to $\mathbf { P } _ { t }$ ) to
+175 prove the contribution of $\mathbf { P } _ { t }$ or reduce the number of parameters. The working program $W _ { t }$ is then
+176 used by the Main Network to execute the input data $x _ { t }$ (see (Fig. 1 (a))). For example, with linear
+177 classifier Main Network, the execution is $y _ { t } = x _ { t } W _ { t }$ . Appendix’s Table 2 summarises the notations
+178 used for important parameters of Neurocoder.
+![](images/f992f503335ad0a3f753e1c8f03e87de2fd352b9c040fd798ec6c09805c37c77.jpg)
+Figure 2: (a) MNIST test set classification error vs the number of steps $( J )$ in Neurocoder (blue), compared with a linear classifier (red). (b) 1st column: Digit images; Middle column: Single-step attention weights for 30 slots in $\mathbf { M } _ { U }$ (vertical axis) for first 3 singular vectors (horizontal axis) for each digit; Last column: Multi-step attention weights for 10 slots in $\mathbf { M } _ { U }$ (vertical axis) for first 3 singular vectors (horizontal axis). Multi-step attention is able to produce far more diverse patterns with fewer slots - 10 slots compared to single-step 30 slots. (c) Two attention patterns of singlestep Neurocoder. The binary decision tree derived from single-step Neurocoder’s attention patterns. The two patterns across components represent the decisions going up and down across the binary tree. Visualisation for (d) multi-step $J = 5$ , 20 memory slots) and (e) single-step $J = 1$ , 10 memory slots) cases showing while processing a sequence of the polynomial auto-regression task. The Neurocoder’s attentions to $\mathbf { M } _ { U }$ that form the first component of the active program are shown over sequence timesteps (upper) with Neurocoder’s $y _ { t }$ prediction (orange) and ground truth (blue) (lower). The vertical dash green lines separate polynomial chunks. Each chuck represents a local pattern, and thus ideally requires a specific active program to compute the input $x _ { t }$ . Although both predict well, only the multi-step Neurocoder discovers the chunk boundaries, assigning program attention to the first component in accordance with sequence changes.
+# 179 3 Results
+180 To demonstrate the flexibility of Neurocoder framework, we consider different learning paradigms:
+181 instance-based, sequential, multi-task and continual learning. We do not focus on breaking perfor
+182 mance records by augmenting state-of-the-art models with Neurocoder. Rather our inquiry is on
+183 re-coding feed-forward layers with the Neurocoder’s programs and testing on varied data types to
+184 demonstrate its intrinsic properties. For some experiments, we include ablation studies.
+185 We compare the performance of diverse Main Networks (MN) with and without Neurocoder. We
+186 also augment the Main Networks with other recent conditional computing methods, either modular
+187 (sparse Mixture of Experts, Neural Stored-program Memory) or monolithic (HyperNets, FiLM) to
+188 form stronger baselines across our experiments. In our experiments, we always apply Neurocoder
+189 to all layers of multi-layer perceptrons (MLP) or just the final feed-forward layer of deep CNN
+190 networks (LeNet, DenseNet, ResNet), RNNs (GRU, LSTM), MANN (NTM). Other competitors
+191 such as MOE, NSM, HyperNet and FiLM are applied to the Main Networks in the same manner.
+![](images/7e25b32d05c9ae7db2df1ea7bbd96b41284179be7b8a5c34b278f599cab1c811.jpg)
+Figure 3: Learning curves (mean and std. over 5 runs) on representative Atari 2600 games. All baselines are applied to the actor/critic networks in the A3C agent.
+# 192 3.1 Instance-based learning - Object Recognition
+We tested Neurocoder on instance-based learning through classical image classification tasks using MNIST [24] and CIFAR [21] datasets. The first experiment interpreted Neurocoder’s behaviour in classifying digits into 10 classes $( 0 - 9 )$ using linear classifier Main Network. With equivalent model size, Neurocoder using the novel recurrent attention surpasses the performance of the linear classifier [24] by up to $5 \%$ (Fig. 2 (a)).
+To differentiate the input, Neurocoder attends to different components of the active program to guide the decision-making process. Fig. 2 (b) shows single-step and multi-step attention to the first 3 singular vectors for each digit across memory slots. Multi-step attention produces richer patterns compared to single-step Neurocoder that manages only 2 attention weight patterns.
+Fig. 2 (c) illustrates how Neurocoder performs modular learning by showing the attention assignment for top 3 singular vectors as a binary decision tree. Digits under the same parental node share similar attention paths, and thereby similar active programs. Some digits look unique (e.g. 7) resulting in active programs composed of unique attention paths, discriminating themselves early in the decision tree. Some digits (e.g. 0 and 9) share the same attention pattern for the first 3 components and are thus unclassifiable. They can only be distinguished by considering more singular vectors.
+We integrated Neurocoder with deep networks - 5-layer LeNet and 100-layer DenseNet - and tested on CIFAR datasets. Neurocoder significantly outperformed the original Main Networks with performance gain $1 - 5 \%$ . Compared with recent conditional computing models such as sparse Mixture of Experts (MOE [35]) and Neural Stored-program Memory (NSM [22]), Neurocoder required a tenth of the number of parameters and performed better by up to $8 - 1 0 \%$ (see Appendix’s Table 3).
+# 3.2 Sequential learning - Adaption to sequence changes and game playing using reinforcement learning
+Recurrent neural networks (RNN) can learn from sequential data by updating the hidden states of the networks. However, this does not suffice when local patterns shift, as is often the case. We now demonstrate that Neurocoder helps RNNs overcome this limitation by composing diverse programs to handle sequence changes.
+Synthetic polynomial auto-regression We created a simple auto-regression task in which data points are sampled from polynomial function chunks that change over time. The Main Network is a strong RNN–Gated Recurrent Unit (GRU [5]). We found that GRU integrated with a single-step or multi-step Neurocoder converged much faster than all other baselines. The other conditional computing counterparts (HyperNet [13], FiLM [28]) adapt by re-scaling weights or activation of the GRU, which were shown inferior to our modular approach (Appendix’s Fig. 6).
+225 Visualising the first singular vector attention weights in $\mathbf { M } _ { U }$ , we find that the multi-step attention
+226 Neurocoder changes its attention following polynomial changes - it attends to the same singular pro
+227 gram when processing data from the same polynomial and alters attention for data from a different
+228 polynomial (Fig. 2(d)). In contrast, the single-step Neurocoder only changes its attention when
+229 there is a remarkable change in $y$ -coordinate values (Fig. 2(e)). Although single-step Neurocoder
+230 converges well, it did not discover the underlying structure of the data, and thus underperformed
+231 the multi-step Neurocoder. We hypothesise that when recurrence is employed, usage-based atten
+232 tion takes effect, stipulating better memory utilisation and diverse attentions over timesteps. We ran
+233 multi-step Neurocoder without usage-based attention. The results were worse than the full multi
+234 step Neurocoder, which confirms our hypothesis (Appendix’s Fig. 6).
+<table><tr><td>Method</td><td>MN (MLP[17])</td><td>MN (MLP ours)</td><td>NSM</td><td>Neurocoder</td></tr><tr><td>Adam</td><td>55.16±1.38</td><td>53.55±1.27</td><td>54.85±2.81</td><td>58.46±0.46</td></tr><tr><td>Adagrad</td><td>58.08±1.06</td><td>57.83±2.74</td><td>58.42±1.87</td><td>62.28±4.03</td></tr><tr><td>L2</td><td>66.00±3.73</td><td>64.37±2.40</td><td>62.83±7.21</td><td>69.89±1.72</td></tr><tr><td>SI</td><td>64.76±3.09</td><td>64.41±3.36</td><td>64.36±2.99</td><td>67.96±3.22</td></tr><tr><td>EWC</td><td>58.85±2.59</td><td>58.41±2.37</td><td>58.12±3.24</td><td>65.66±1.25</td></tr><tr><td>O-EWC</td><td>57.33±1.44</td><td>57.78±1.84</td><td>58.55±3.40</td><td>73.97±1.50</td></tr></table>
+Table 1: Incremental domain continual learning with Split MNIST. Final test accuracy (mean and std.) over 10 runs.
+Atari game reinforcement learning We used reinforcement learning as a further testbed to show the ability to adapt to environmental changes. We performed experiments on several Atari 2600 games [3] wherein the agent was implemented as the Asynchronous Advantage Actor-Critic (A3C [26]). In the Atari platform, agents are allowed to observe the screen snapshot of the games and act to earn the highest score. We augmented the A3C by employing Neurocoder’s working programs for feed-forward layers of the actor and critic networks, aiming to decompose the policy and value function into singular programs that were selected depending on the game state.
+242 Frostbite and Montezuma’s Revenge. These games are known to be challenging for A3C and other
+243 algorithms [26]. We trained A3C and HyperNet-based A3C for over 300 million steps, yet these
+244 models did not show any sign of learning, performing equivalently to random agents. For such com
+245 plicated environments with sparse rewards, both the monolithic neural networks and the HyperNet’s
+246 unstored fast-weights fail to learn (almost zero scores). In contrast, Neurocoder enabled A3C to
+247 achieve from 1, 500 to 3, 000 scores on these environments (Fig. 3), confirming the importance of
+248 decomposing a complex solution to smaller, simple stored programs.
+# 49 3.3 Multi-task learning - Solving mutliple algorithms simultenously
+Here we explore the modular learning capability of Neurocoder in multi-task setting. Inspired by algorithmic sequencing tasks [22], we created a challenging sequential multi-task benchmark wherein the input sequence is a series of sub-sequences from 4 algorithms: Copy, Repeat Copy, Associative Recall and Priority Sort [11]. Each sub-sequence, following a task identification vector, represents the input for each task. In each input sequence, $n$ tasks were sampled from the set of 4 algorithms randomly with replacement and the output sequences were created correspondingly.
+We trained a MANN–Neural Turing Machine (NTM [11]) Main Network with FiLM, HyperNet and our Neurocoder augmentation on sequences of $n = 4$ tasks, and tested with sequences of $n = 4$ and $n = 8$ tasks. Appendix’s Fig. 7 demonstrates that Neurocoder was performant in both test settings, not only achieving lowest error on $n = 4$ , but also being the only one generalised well to $n = 8$ scenario, which was unseen during training.
+# 3.4 Continual learning $-$ Learning tasks sequentially without catastrophic forgetting
+In continual learning, standard neural networks often suffer from “catastrophic forgetting” in which they cannot retain knowledge acquired from old tasks upon learning new ones [10]. Our Neurocoder offers natural mitigation of such catastrophic forgetting in neural networks by attending to different singular programs whilst learning different tasks.
+In this case, in addition to the Main Network, we examine several continual learning algorithms with and without Neurocoder. These algorithms, including Elastic Weight Consolidation (EWC [41]) and Synaptic Intelligence (SI [41]), work by regularising the loss function and thus can be easily combined with Neurocoder by modifying the loss $\mathcal { L } _ { t a s k }$ . We demonstrate that Neurocoder
+70 can improve these continual learning algorithms without requiring additional assumptions as in other
+1 approaches [25, 36, 34] that either utilise task embedding or replay memory.
+Split MNIST We first considered the split MNIST dataset–a standard continual learning benchmark wherein the original MNIST was split into a 5 2-way classification tasks, consecutively presented to a Multi-layer Perceptron Main Network (MLP). We followed the benchmarking as in [17] in which various optimisers and state-of-the-art continual learning methods were examined under incremental task and domain scenarios. We measured the performance of the MLP versus Neurocoder and NSM under each continual learning method. In both scenarios, Neurocoder was compatible with all continual leaning methods, demonstrating superior performance over MLP and NSM with performance gain between 1 to $1 6 \%$ (see Appendix’s Table 5 and 1).
+280 Split CIFAR We verified the scalability of Neurocoder to more challenging datasets. We split
+281 CIFAR datasets as in the split MNIST, resulting in 5-task 2-way split CIFAR10 and a 20-task 5-way
+282 split CIFAR100. We used Main Network ResNet [15]–a very deep CNN architecture.
+283 When we stressed the orthogonal loss $a = 1 0$ ) and used bigger program memory (100 slots), Neu
+284 rocoder improved ResNet classification by $1 5 \%$ and $1 0 \%$ on CIFAR10 and CIFAR100, respectively.
+285 When we integrated Neurocoder with Synaptic Intelligence (SI [41]), the performance was further
+286 improved, maintaining a stable performance above $8 0 \%$ accuracy for CIFAR10 and outperforming
+287 using SI alone by $1 0 \%$ for CIFAR100 (see Appendix’s Fig. 8).
+# 288 4 Discussion
+Our experiments demonstrate that Neurocoder is capable of re-coding Neural Programs in distinctive neural networks, amplifying their capabilities in diverse learning scenarios: instance-based, sequential, multi-task and continual learning. This consistently results in significant performance increase, and further creates novel robustness to pattern shift and catastrophic forgetting. This ability for each architecture to re-code itself is made possible without changing the way it is trained, or majorly increasing the number of parameters it needs to learn (see Appendix Table 7).
+The MNIST problem illustrates the reasoning process of Neurocoder when classifying digit images wherein its singular program assignment resembles a binary tree decision-making process - it shows how some singular programs are shared, others are not. The polynomial auto-regression problem highlights the importance of efficient memory utilisation in re-constructing the working program enabling discovery of hidden structures in sequential data. Training our framework with reinforcement learning, we enable neural agents to solve complex games wherein traditional methods fail or learn slowly. Neurocoder also works well with multi-task setting, as shown in the challenging multi-algorithm benchmark. Finally, continual learning problems show that Neurocoder mitigates catastrophic forgetting efficiently under different learning settings/algorithms.
+304 Our solution offers a single framework that is scalable and adaptable to various problems and learn
+305 ing paradigms. Unlike previous attempts to employ a bank of separate big programs [20, 35, 22],
+306 Neurocoder maintains only shareable, smaller components that can reconstruct the whole program
+307 space, thereby heavily utilising the parameters and preventing the model from proliferating. We
+308 note that Neurocoder is orthogonal to approaches employing tensor decomposition to reduce the
+309 number of parameters or hasten the computation [27, 23]. Neurocoder composes rather than decom
+310 pose the neural weights. Our aim is not only to enable efficient parameter usage, but also achieve
+311 general-purpose computing power, outperforming other methods in numerous learning problems.
+312 One limitation of this work is the number of additional hyperparameters, which prevents us from
+313 fully tuning Neurocoder. Our research aims to add new capabilities to current neural networks to
+314 improve their performance and make them robust in different learning scenario. Hence, we do
+315 not see any intermediate negative societal impact. In future work, we will extend Neurocoder’s
+316 application beyond feed-forward layers. It would be interesting to efficiently replace all neural layers
+317 including CNN or Transformer by Neurocoder’s programs. We can also further extend Neurocoder’s
+318 ability by allowing a growing Program Memory, in which the model decides to add or erase memory
+319 slots as the number of data patterns grows or shrinks beyond the current program space’s capacity.
+320 Such a system represents a more flexible general-purpose computer that can dynamically allocate
+321 computing resources by itself without human pre-specification.
+References
+[1] Jacob Andreas, Marcus Rohrbach, Trevor Darrell, and Dan Klein. Neural module networks. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 39–48, 2016.
+[2] Dzmitry Bahdanau, Kyunghyun Cho, and Yoshua Bengio. Neural machine translation by jointly learning to align and translate. In International Conference on Learning Representations, 2015.
+[3] Marc G Bellemare, Yavar Naddaf, Joel Veness, and Michael Bowling. The arcade learning environment: An evaluation platform for general agents. Journal of Artificial Intelligence Research, 47:253–279, 2013.
+[4] Yoshua Bengio, Nicholas Léonard, and Aaron Courville. Estimating or propagating gradients through stochastic neurons for conditional computation. arXiv preprint arXiv:1308.3432, 2013.
+[5] Kyunghyun Cho, Bart van Merriënboer, Caglar Gulcehre, Dzmitry Bahdanau, Fethi Bougares, Holger Schwenk, and Yoshua Bengio. Learning phrase representations using RNN encoder– decoder for statistical machine translation. In Conference on Empirical Methods in Natural Language Processing (EMNLP), pages 1724–1734. Association for Computational Linguistics, October 2014.
+[6] JC Eccles. The modular operation of the cerebral neocortex considered as the material basis of mental events. Neuroscience, 6(10):1839–1855, 1981.
+[7] Gerald M Edelman. Neural darwinism: selection and reentrant signaling in higher brain function. Neuron, 10(2):115–125, 1993.
+[8] Gerald M Edelman and Vernon B Mountcastle. The mindful brain: cortical organization and the group-selective theory of higher brain function. Massachusetts Inst of Technology Pr, 1978.
+[9] Richard SJ Frackowiak. Human brain function. Elsevier, 2004.
+[10] Robert M French. Catastrophic forgetting in connectionist networks. Trends in cognitive sciences, 3(4):128–135, 1999.
+[11] Alex Graves, Greg Wayne, and Ivo Danihelka. Neural turing machines. arXiv preprint arXiv:1410.5401, 2014.
+[12] Alex Graves, Greg Wayne, Malcolm Reynolds, Tim Harley, Ivo Danihelka, Agnieszka Grabska-Barwinska, Sergio Gómez Colmenarejo, Edward Grefenstette, Tiago Ramalho, John ´ Agapiou, et al. Hybrid computing using a neural network with dynamic external memory. Nature, 538(7626):471–476, 2016.
+[13] David Ha, Andrew M. Dai, and Quoc V. Le. Hypernetworks. In International Conference on Learning Representations, 2017.
+[14] Bart LM Happel and Jacob MJ Murre. Design and evolution of modular neural network architectures. Neural networks, 7(6-7):985–1004, 1994.
+[15] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 770–778, 2016.
+[16] Sepp Hochreiter and Jürgen Schmidhuber. Long short-term memory. Neural computation, 9(8):1735–1780, 1997.
+[17] Yen-Chang Hsu, Yen-Cheng Liu, Anita Ramasamy, and Zsolt Kira. Re-evaluating continual learning scenarios: A categorization and case for strong baselines. In NeurIPS Continual learning Workshop, 2018.
+[18] Gao Huang, Zhuang Liu, Laurens Van Der Maaten, and Kilian Q Weinberger. Densely connected convolutional networks. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 4700–4708, 2017.
+[19] D.H. Hubel. Eye, Brain, and Vision. Scientific American Library series. Scientific American Library, 1988.
+[20] Robert A Jacobs, Michael I Jordan, Steven J Nowlan, and Geoffrey E Hinton. Adaptive mixtures of local experts. Neural computation, 3(1):79–87, 1991.
+[21] Alex Krizhevsky, Geoffrey Hinton, et al. Learning multiple layers of features from tiny images. TR-2009, 2009.
+[22] Hung Le, Truyen Tran, and Svetha Venkatesh. Neural stored-program memory. In International Conference on Learning Representations, 2020.
+[23] V Lebedev, Y Ganin, M Rakhuba, I Oseledets, and V Lempitsky. Speeding-up convolutional neural networks using fine-tuned cp-decomposition. In International Conference on Learning Representations, 2015.
+[24] Yann LeCun, Léon Bottou, Yoshua Bengio, and Patrick Haffner. Gradient-based learning applied to document recognition. Proceedings of the IEEE, 86(11):2278–2324, 1998.
+[25] David Lopez-Paz and Marc’Aurelio Ranzato. Gradient episodic memory for continual learning. In Advances in neural information processing systems, pages 6467–6476, 2017.
+[26] Volodymyr Mnih, Adria Puigdomenech Badia, Mehdi Mirza, Alex Graves, Timothy Lillicrap, Tim Harley, David Silver, and Koray Kavukcuoglu. Asynchronous methods for deep reinforcement learning. In International conference on machine learning, pages 1928–1937, 2016.
+[27] Alexander Novikov, Dmitrii Podoprikhin, Anton Osokin, and Dmitry P Vetrov. Tensorizing neural networks. In Advances in neural information processing systems, pages 442–450, 2015.
+[28] Ethan Perez, Florian Strub, Harm De Vries, Vincent Dumoulin, and Aaron Courville. FiLM: Visual Reasoning with a General Conditioning Layer. In AAAI Conference on Artificial Intelligence, New Orleans, United States, February 2018.
+[29] Clemens Rosenbaum, Tim Klinger, and Matthew Riemer. Routing networks: Adaptive selection of non-linear functions for multi-task learning. In International Conference on Learning Representations, 2018.
+[30] David E Rumelhart, Geoffrey E Hinton, and Ronald J Williams. Learning representations by back-propagating errors. Nature, 323(6088):533–536, 1986.
+[31] Adam Santoro, Sergey Bartunov, Matthew Botvinick, Daan Wierstra, and Timothy Lillicrap. Meta-learning with memory-augmented neural networks. In International conference on machine learning, pages 1842–1850, 2016.
+[32] Jürgen Schmidhuber. Making the world differentiable: On using self-supervised fully recurrent neural networks for dynamic reinforcement learning and planning in non-stationary environm nts. TR FKI-126-90, 1990.
+[33] Jürgen Schmidhuber. Learning to control fast-weight memories: An alternative to dynamic recurrent networks. Neural Computation, 4(1):131–139, 1992.
+[34] Joan Serra, Didac Suris, Marius Miron, and Alexandros Karatzoglou. Overcoming catastrophic forgetting with hard attention to the task. In International Conference on Machine Learning, pages 4548–4557, 2018.
+[35] Noam Shazeer, Azalia Mirhoseini, Krzysztof Maziarz, Andy Davis, Quoc V. Le, Geoffrey E. Hinton, and Jeff Dean. Outrageously large neural networks: The sparsely-gated mixture-ofexperts layer. In International Conference on Learning Representations, 2017.
+[36] Hanul Shin, Jung Kwon Lee, Jaehong Kim, and Jiwon Kim. Continual learning with deep generative replay. In Advances in Neural Information Processing Systems, pages 2990–2999, 2017.
+[37] A.M Turing. On computable numbers, with an application to the entscheidungsproblem. In Proceedings of the London Mathematical Society, 1936.
+[38] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin. Attention is all you need. In Advances in neural information processing systems, pages 5998–6008, 2017.
+[39] Christoph von der Malsburg. The correlation theory of brain function, 1981.
+[40] John Von Neumann. First draft of a report on the edvac. IEEE Annals of the History of Computing, 15(4):27–75, 1993.
+[41] Friedemann Zenke, Ben Poole, and Surya Ganguli. Continual learning through synaptic intelligence. Proceedings of machine learning research, 70:3987, 2017.
+1. For all authors...
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]
+(b) Did you describe the limitations of your work? [Yes] See Discussion and Appendix’s "Training procedure and hyper-parameter selections."
+(c) Did you discuss any potential negative societal impacts of your work? [Yes] See Discussion.
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
+2. If you are including theoretical results...
+(a) Did you state the full set of assumptions of all theoretical results? [N/A] (b) Did you include complete proofs of all theoretical results? [N/A]
+3. If you ran experiments...
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [No] Data is public, provided with link. Code will be avaialble after published. All training details are available and can be used to implement and reproduce the results.
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] See Appendix’s "Training procedure and hyper-parameter selections."
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] See Experimental Results.
+(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] See Appendix’s "Training procedure and hyper-parameter selections.
+4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
+(a) If your work uses existing assets, did you cite the creators? [Yes]
+(b) Did you mention the license of the assets? [No] All assets are public. We will mention the license detail after the paper is published.
+(c) Did you include any new assets either in the supplemental material or as a URL? [No]
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [No]
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [No]
+5. If you used crowdsourcing or conducted research with human subjects...
+(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]
+(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]
+(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]

parse/train/O0ow9RQW1nP/O0ow9RQW1nP_content_list.json ADDED Viewed

	@@ -0,0 +1,1216 @@

+[
+    {
+        "type": "text",
+        "text": "Neurocoder: Learning General-Purpose Computation Using Stored Neural Programs ",
+        "text_level": 1,
+        "bbox": [
+            228,
+            122,
+            769,
+            172
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous Author(s)   \nAffiliation   \nAddress   \nemail ",
+        "bbox": [
+            423,
+            226,
+            578,
+            281
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
+            462,
+            318,
+            535,
+            334
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Artificial Neural Networks are functionally equivalent to special-purpose computers. Their inter-neuronal connection weights represent the learnt Neural Program that instructs the networks on how to compute the data. However, without storing Neural Programs, they are restricted to only one, overwriting learnt programs when trained on new data. Here we design Neurocoder, a new class of generalpurpose neural networks in which the neural network “codes” itself in a dataresponsive way by composing relevant programs from a set of shareable, modular programs stored in external memory. For the first time, a Neural Program is efficiently treated as a datum in memory. Integrating Neurocoder into current neural architectures, we demonstrate new capacity to learn modular programs, reuse simple programs to build complex ones, handle pattern shifts and remember old programs as new ones are learnt, and show substantial performance improvement in solving object recognition, playing video games and continual learning tasks. ",
+        "bbox": [
+            148,
+            348,
+            766,
+            529
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "14 1 Introduction ",
+        "text_level": 1,
+        "bbox": [
+            148,
+            553,
+            310,
+            570
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "15 From its inception in 1943 until recently, the fundamental architectures of Artificial Neural Net  \n16 works remained largely unchanged - a program is executed by passing data through a network of   \n17 artificial neurons whose inter-neuronal connection weights are learnt through training with data.   \n18 These inter-neuronal connection weights, or Neural Programs, correspond to a program in modern   \n19 computers [32]. Memory Augmented Neural Networks (MANN) are an innovative solution allow  \n20 ing networks to access external memory for manipulating data [11, 12]. But they were still unable   \n21 to store Neural Programs in such external memory, and this severely limits machine learning. Stor  \n22 ing inter-neuronal connection weights only in their network does not permit modular separation of   \n23 Neural programs and is analogous to a computer with one fixed program. Recent works introduce   \n24 conditional computation via adjusting or activating parts of a network in an input-dependent manner   \n25 [39, 33, 4, 13, 28], but networks remain monolithic. Current networks forget when retrained, old   \n26 inter-neuronal connection weights are merged with new ones or erased.   \n27 The brain is modular, not a monolithic system [8, 6]. Neuroscience research indicates that the brain is   \n28 divided into functional modules [19, 7, 9]. If the neural program for each module is kept in separate   \n29 networks, networks proliferate. Modular neural networks, another form of conditional computation,   \n30 combine the output of multiple expert networks, but as the experts grow, the networks grow drasti  \n31 cally [20, 14, 35, 29]. This requires huge computational storage and introduces redundancy as these   \n32 experts do not share common basic programs.   \n33 A pathway out of this bind is to keep such basic programs in memory and combine them as required.   \n34 This brings neural networks towards modern general-purpose computers that use the stored-program   \n35 principle [37, 40] to efficiently access reusable programs in external memory. Here we show how   \n36 Neurocoder, a new neural framework, introduces a new class of general-purpose conditional compu  \n37 tation machines in which a neural network can be “coded” in an input-dependent manner. Efficient   \n38 decomposition of Neural Programs creates shareable modular components that can reconstruct the   \n39 whole program space. These components change their “shapes” based on training and are stored   \n40 in an external Program Memory. Then, in a data-responsive way, a Program Controller retrieves   \n41 relevant components to build the Neural Program. This is analogous to shape-shifting Lego bricks   \n42 that can be reused to build unlimited shapes and structures (See Appendix Fig. 4).   \n43 Using adaptive modular components vastly increases the learning capacity of the neural network   \n44 by allowing re-utilisation of parameters, effectively curbing network growth as programs increase.   \n45 More importantly, unlike pre-defined sub-networks or modules [20, 1] that combine at activation   \n46 level, the construction of our modular components is dynamic and performed on the weight space.   \n47 The Neural Program construction is learnt through training via traditional backpropagation [30] as   \n48 the architecture is end-to-end differentiable. ",
+        "bbox": [
+            147,
+            584,
+            825,
+            751
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            147,
+            756,
+            825,
+            840
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            147,
+            845,
+            823,
+            901
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "image",
+        "img_path": "images/e389c763e9b4533f0b062c87f24ebd5da2780f7e18d1c6818a55a57ce1880a0c.jpg",
+        "image_caption": [
+            "Figure 1: Neurocoder (a) The Main Network uses a working program to compute the output for the input. Here only the final layer of the Main Network is adaptively loaded with the working program $( I )$ . Other layers use traditional Neural Programs as connection weights (fixed-after-training). (b) The Program Controller’s composition network controls access to the Program Memory, emitting queries and interpolating gate control signals in response to the input (2). It then performs recurrent multi-head program attention to the Program Status (3), triggering attention weights to the Singular Programs (4). The attended Singular Programs form an active program using low-rank approximation (5). Residual program produced by the Program Controller’s integration network (6) plus the active program derives the working program. (c) The Program Memory stores the representations (singular programs) required to reconstruct the active program to be used by the Program Controller. Access is controlled through the Program Status including keys $( k )$ , and slot usage $( m )$ that are updated during the training and computation (7). "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            186,
+            88,
+            818,
+            367
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            147,
+            578,
+            825,
+            661
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            147,
+            667,
+            825,
+            750
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "49 2 Methods ",
+        "text_level": 1,
+        "bbox": [
+            155,
+            777,
+            277,
+            794
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "2.1 System overview ",
+        "text_level": 1,
+        "bbox": [
+            174,
+            814,
+            326,
+            828
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "51 A Neurocoder is a neural network (Main Network) coupled to an external Program Memory through   \n52 a Program Controller. The working program of the Main Network processes the input data to pro  \n53 duce the output. This working program is “coded” by the Program Controller by creating an input  \n54 dependent active program from the Program Memory (Fig. 1). The following gives a high-level   \n55 description of the Neurocoder framework and then the details. ",
+        "bbox": [
+            147,
+            842,
+            823,
+            911
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "56 Neurocoder stores Singular Value Decomposition of Neural Programs in Program Memory ",
+        "text_level": 1,
+        "bbox": [
+            150,
+            90,
+            805,
+            106
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "57 The Neural Program needs to be stored efficiently in Program Memory. This is challenging as there   \n58 may be millions of inter-neuronal connection weights, thus storing them directly ([22]) is grossly   \n59 inefficient. Instead, the Neurocoder forms the basis of a subspace spanned by Neural Programs and   \n60 stores the singular values and vectors of this subspace in memory slots of the Program Memory   \n61 (hereafter referred to as singular programs). Based on the input, relevant singular programs are   \n62 retrieved, a new program is reconstructed and then loaded in the Main Network to process the input.   \n63 This representational choice significantly reduces the number of stored elements and allows each   \n64 singular program to effectively represent a unitary function of the active program. ",
+        "bbox": [
+            145,
+            114,
+            825,
+            227
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "65 The active program matrix $\\mathbf { P }$ can be composed by standard low-rank approximation as ",
+        "bbox": [
+            147,
+            231,
+            743,
+            247
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/9f81cb11dec9c11b78cca6d7f7e9e1a93d9f3d74864311254010fb6362f61e22.jpg",
+        "text": "$$\n\\mathbf { P } = \\mathbf { U S V } ^ { \\mathbf { T } } = \\sum _ { n } ^ { r _ { m } } \\sigma _ { n } u _ { n } v _ { n } ^ { \\top }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            401,
+            252,
+            594,
+            292
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "66 where $\\mathbf { U }$ and $\\mathbf { V }$ are matrices of the left and right singular vectors, and S the matrix of singular values.   \n67 $r _ { m }$ is the total number of components we want to retrieve. $\\left\\{ \\sigma _ { n } \\right\\} _ { n = 1 } ^ { r _ { m } }$ is the attended singular values,   \n68 {un} mn= and $\\{ v _ { n } \\} _ { n = 1 } ^ { r _ { m } }$ the attended singular vectors of S, U, and $\\mathbf { V }$ , respectively. The Program   \n69 Memory is crafted as three singular program memories $\\{ { \\bf M } _ { U } , { \\bf M } _ { V } , { \\bf M } _ { S } \\}$ –each of their memory   \n70 slot stores a singular component or singular program. The process “codes” the active program   \n71 using singular programs from the program memories. The coding is conditioned on input $x _ { t }$ , yet   \n72 we drop index $t$ for notation simplification and leave the details on the computation of $\\sigma _ { n } , u _ { n } , v _ { n }$ in   \n73 Sec. 2.2.   \n74 The Program Memory also maintains the status for each singular program in terms of access and   \n75 usage. To access a singular program, program keys $( k )$ are used. These keys are low-dimensional   \n76 vectors that represent the singular program function and computed by a neural network that ef  \n77 fectively compresses the singular program. The program usage $( m )$ measures memory utilisation,   \n78 recording how much a memory slot is used in constructing a program. The components of the   \n79 Program Memory are summarised in Fig. 1 (c).   \n80 Recurrent multi-head program attention mechanisms for program storage and retrieval   \n81 Neural networks use the concept of differentiable attention to access memory [11, 2]. This de  \n82 fines a weighting distribution over the memory slots essentially weighting the degree to which each   \n83 memory slot participates in a read or write operation. This is unlike conventional computers that use   \n84 a unique address to access a single memory slot.   \n85 Here we use two kinds of attention. First is content-based attention [11, 12] to ensure that the singu  \n86 lar program is selected based on its functionality and the data input. This is achieved by producing   \n87 a query vector based on the input and comparing it to the program keys $( k )$ using cosine similarity.   \n88 Higher cosine similarity scores indicate higher attention weights to the singular programs associated   \n89 with those program keys. Second, to encourage better memory utilisation, higher attention weights   \n90 are assigned to slots with lower program usage $( m )$ through usage-based attention [12, 31]. The   \n91 attention weights from the two schemas are then combined using interpolating gates to compose the   \n92 final attention weights to the Program Memory.   \n93 We adapt multi-head attention [11, 38] that applies multiple attentions in parallel to retrieve $H$ singu  \n94 lar components. Besides, we introduce a recurrent attention mechanism, in which multi-head access   \n95 is performed recurrently in $J$ steps. The $j$ -th set of $H$ retrieved components is conditioned on the   \n96 previous ones. This recurrent, multi-head attention allows the composition network to incrementally   \n97 search for optimal components for building relevant active programs. ",
+        "bbox": [
+            147,
+            296,
+            825,
+            410
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            147,
+            416,
+            825,
+            500
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            147,
+            513,
+            825,
+            583
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            145,
+            589,
+            825,
+            700
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            147,
+            707,
+            823,
+            776
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "98 Neurocoder learns to “code” a relevant working program via training ",
+        "text_level": 1,
+        "bbox": [
+            150,
+            790,
+            658,
+            805
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "99 The structure of the Program Memory and the role of the Program Controller facilitates the au  \n100 tomatic construction of working programs via training. The Program Controller controls memory   \n101 access through its composition network that creates the attention weight defining how to weight the   \n102 singular programs in the memories. A weighted summation of the singular programs results in the   \n103 attended singular program. Applying the recurrent multi-head attention described earlier, multiple   \n104 attended singular programs are retrieved to construct an active program (Eq. 1). Then the Program   \n105 Controller generates a residual program using its integration network, adding to the active program   \n106 to produce the working program of the Main Network. This addition enables creation of flexible   \n107 higher-rank working programs, which compensates for the low-rank coding process. The structure   \n108 of the Program Controller is illustrated in Fig. 1 (b).   \n109 The singular programs are trained to represent unitary functions necessary for any computation   \n110 whilst the composition and integration networks are trained to compose the relevant programs for   \n111 the considering task. As such, beside minimising the task loss, we enforce orthogonality of stored   \n112 singular vectors by minimising $\\mathcal { L } _ { o } = \\mathbf { M } _ { U } \\mathbf { M } _ { U } ^ { \\top } - \\mathbf { \\bar { I } } + \\mathbf { M } _ { V } \\mathbf { M } _ { V } ^ { \\top } - \\mathbf { I }$ . The parameters of the networks,   \n113 and the stored singular programs are adjusted using gradient training via minimising the total loss ",
+        "bbox": [
+            143,
+            814,
+            825,
+            911
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            142,
+            90,
+            826,
+            133
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            140,
+            138,
+            825,
+            210
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/73ae3d1e18ef6e591429508b2ed4e92a9f313315a3a72f46d0c305e7d5f3e144.jpg",
+        "text": "$$\n\\mathscr { L } = \\mathcal { L } _ { t a s k } + a \\mathcal { L } _ { o }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            437,
+            228,
+            560,
+            243
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "where $\\mathcal { L } _ { t a s k }$ represents the supervised task loss and $\\mathcal { L } _ { o }$ represents the orthogonal loss weighted by a hyper-parameter $a$ to enforce orthogonality of the singular vectors. ",
+        "bbox": [
+            161,
+            244,
+            825,
+            273
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "2.2 Attention mechanisms for Program Memory ",
+        "text_level": 1,
+        "bbox": [
+            160,
+            287,
+            522,
+            303
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "117 Here we ddenoted as $w _ { i n } ^ { u , v , \\sigma }$ progra)–the a attention mechanintion weight to the $i$ ms used in this paper. Given -th slot of the singular progra $w _ { i n } ^ { u }$ , e $w _ { i n } ^ { v }$ , i $w _ { i n } ^ { \\sigma }$ $\\mathbf { M } _ { U }$ o, $\\mathbf { M } _ { V }$   \n119 , we retrieve the $n$ -th singular vector as follows, ",
+        "bbox": [
+            142,
+            313,
+            825,
+            356
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7f593b14fbe5fa2db83bfae351abab39af3ebcf4aa3fc552641badb79ceb64dc.jpg",
+        "text": "$$\nu _ { n } = \\sum _ { i = 1 } ^ { P _ { u } } w _ { i n } ^ { u } \\mathbf { M } _ { U } \\left( i \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
+            426,
+            371,
+            571,
+            415
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/22ef38c2bf214494039065f6b7c509007ff039d38b1ab448d686f393e65d8af4.jpg",
+        "text": "$$\nv _ { n } = \\sum _ { i = 1 } ^ { P _ { v } } w _ { i n } ^ { v } \\mathbf { M } _ { V } \\left( i \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
+            426,
+            421,
+            570,
+            464
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "120 For the singular values, we need to enforce $\\sigma _ { 1 } > \\sigma _ { 2 } > . . . > \\sigma _ { r _ { m } } > 0$ , thus we retrieve using ",
+        "bbox": [
+            138,
+            463,
+            781,
+            479
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/189a9f5c9f12697e6d17a3e2309cbe37090458ece1e4757b8e6d2895b27d7fbc.jpg",
+        "text": "$$\n\\sigma _ { n } = \\left\\{ \\begin{array} { l l } { \\mathrm { s o f t p l u s } \\left( \\sum _ { i = 1 } ^ { P _ { s } } w _ { i n } ^ { \\sigma } \\mathbf { M } _ { S } \\left( i \\right) \\right) } & { n = r _ { m } } \\\\ { \\sigma _ { n + 1 } + \\mathrm { s o f t p l u s } \\left( \\sum _ { i = 1 } ^ { P _ { s } } w _ { i n } ^ { \\sigma } \\mathbf { M } _ { S } \\left( i \\right) \\right) } & { n < r _ { m } } \\end{array} \\right.\n$$",
+        "text_format": "latex",
+        "bbox": [
+            310,
+            493,
+            686,
+            542
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "121 Here, $P _ { u } , P _ { v }$ and $P _ { s }$ are the number of memory slots of $\\mathbf { M } _ { U }$ , $\\mathbf { M } _ { V }$ and $\\mathbf { M } _ { S }$ , respectively. In this   \n122 paper, we set $P = P _ { u } = P _ { v } = P _ { s }$ as the number of memory slots of the Program Memory. We note   \n123 that thead ese notations are speci, and an attention step ed for some data input , hence the full notatio $x _ { t }$ and the hould be $n$ later maps to an attention To simplify notations, we   \n$h$ $j$ $w _ { t i j h } ^ { u , v , \\sigma }$   \n125 will drop $u , v , \\sigma$ from now and describe the computation of a representative $w _ { t i j h }$ for any of the   \n126 three program memories in the following parts. ",
+        "bbox": [
+            142,
+            542,
+            826,
+            627
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "127 Recurrent Access to the Program Memory via the composition network ",
+        "text_level": 1,
+        "bbox": [
+            150,
+            640,
+            671,
+            655
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "128 To perform program attention, the Program Controller employs a composition network (denoted   \n129 as $f _ { \\theta } )$ ), which takes the current input $x _ { t }$ and produce program composition control signals $( \\pmb { \\xi } _ { t } ^ { p } )$ .   \n130 If $f _ { \\theta }$ performs all attentions concurrently via multi-head attention as in [11, 38], it may lead to   \n131 program collapse [22]. To have a better control of the component formation and alleviate program   \n132 collapse, we propose to recurrently attend to the program memory. To this end, we implement $f _ { \\theta }$ as   \n133 $\\pmb { \\xi } _ { t } ^ { p } = \\left\\{ \\pmb { \\xi } _ { t j } ^ { p } \\right\\} _ { j = 1 } ^ { J }$ al network (LST. At access step $j$ M [16]) and let it access the program memory , the recurrent network updates its hidden sta $J$ times, resultin and generates $\\xi _ { t j } ^ { p }$   \n135 using recurrent dynamics as ",
+        "bbox": [
+            140,
+            662,
+            825,
+            782
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8d5ff96f4903eab24618ce065adce652211fa542cb32b96df1282f89efe4a773.jpg",
+        "text": "$$\n{ \\pmb { \\xi } } _ { t j } ^ { p } , h _ { j } = f _ { \\theta } \\left( x _ { t } , h _ { j - 1 } \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
+            419,
+            801,
+            576,
+            819
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "136 where $h _ { 0 }$ is initialized as zeros and $\\xi _ { t j } ^ { p }$ is the program composition control signal at step $j$ that   \n137 depends on both on the input data $x _ { t }$ and the the previous state $h _ { j - 1 }$ . Particularly, the control signal   \n138 contains the queries and the interpolation gates for each head to compute the program attention   \n139 weight: $\\pmb { \\xi } _ { t j } ^ { p } = \\{ q _ { t j h } , g _ { t i j h } \\} _ { h = 1 } ^ { H }$ . Here, at each attention step, we perform multi-head attention with   \n140 $H$ as the number of attention heads and thus, each $\\xi _ { t j } ^ { p }$ consists of $H$ pairs of queries and gates.   \n141 Hence, the total number of retrieved components $\\boldsymbol { r } _ { m } = \\boldsymbol { \\bar { J } } \\times \\boldsymbol { H }$ and the index $n = j \\times H + h$ . ",
+        "bbox": [
+            140,
+            820,
+            825,
+            912
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "142 Attending to Programs by “Name” ",
+        "text_level": 1,
+        "bbox": [
+            143,
+            92,
+            416,
+            106
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "143 Inspired by the content-based attention mechanism for data memory [11], we use the query to look   \n144 for the singular programs. In computer programming, to find the appropriate program for some   \n145 computation, we often refer to the program description or at least the name of the program. Here, we   \n146 create the “name” for our neural programs by compressing the program content to a low-dimensional   \n147 key vector. As such, we employ a neural network $( f _ { \\varphi } )$ to compute the program memory keys as ",
+        "bbox": [
+            140,
+            114,
+            825,
+            185
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/fbbd4d53165389137c65107dbb9b892e43aaf951aa496a542b7ddfbe0a58479f.jpg",
+        "text": "$$\nk _ { i } = f _ { \\varphi } \\left( \\mathbf { M } \\left( i \\right) \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
+            442,
+            203,
+            553,
+            220
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "148 where $\\boldsymbol { k } _ { i } \\in \\mathbb { R } ^ { K }$ and $i$ is the row index of the program memory. Here, $f _ { \\varphi }$ learns to compress each   \n149 memory slot into a $K$ -dimensional vector. As the singular programs evolve, their keys get updated.   \n150 In this paper, we update the program keys after each learning iteration during training.   \n151 Finally the content-based program memory attention $c _ { t i j h }$ is computed using cosine distance be  \n152 tween the program keys $k _ { i }$ and the queries $q _ { t j h }$ as ",
+        "bbox": [
+            140,
+            224,
+            826,
+            267
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            143,
+            273,
+            826,
+            303
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1bb5f46999968c3a1c30f85f187deb3f1106cda326578faa900fb8408b86eb63.jpg",
+        "text": "$$\nc _ { t i j h } = \\mathrm { s o f t m a x } ^ { ( i ) } \\left( \\frac { q _ { t j h } \\cdot k _ { i } } { \\left| \\left| q _ { t j h } \\right| \\right| \\cdot \\left| \\left| k _ { i } \\right| \\right| } \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
+            375,
+            318,
+            622,
+            353
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "153 Making Every Program Count ",
+        "text_level": 1,
+        "bbox": [
+            143,
+            364,
+            390,
+            381
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "154 Similarly to [12, 31], in addition to the content-based attention, we employ a least-used reading   \n155 strategy to encourage the Program Controller to assign different singular programs to different com  \n156 ponents. In particular, we calculate the memory usage for each program slot across attentions as   \n157 ",
+        "bbox": [
+            142,
+            388,
+            825,
+            440
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/20d2e1b3e95cc99e60d768bc0080c3996ae302e11b21cde534c96b944ec0386e.jpg",
+        "text": "$$\nm _ { t i j h } = \\operatorname* { m a x } _ { \\tilde { j } \\leq j } \\left( w _ { t i \\tilde { j } h } \\right)\n$$",
+        "text_format": "latex",
+        "bbox": [
+            423,
+            440,
+            575,
+            470
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "158 Since we want to consider only $l _ { I }$ amongst $P$ memory slots that have smallest usages, let $\\hat { m } _ { t j h } ^ { l _ { I } }$   \n159 denote the value of the $l _ { I }$ -th smallest usage, then the least-used attention is computed as ",
+        "bbox": [
+            143,
+            477,
+            825,
+            508
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3eff87705475f2c68a1c4480bd1c57076406de4a285ac39dac6df397bafd8964.jpg",
+        "text": "$$\nl _ { t i j h } = \\left\\{ \\begin{array} { l l } { \\underset { i } { \\operatorname* { m a x } } \\left( m _ { t i j h } \\right) - m _ { t i j h } } & { ; m _ { t i j h } \\leq \\hat { m } _ { t j h } ^ { l _ { I } } } \\\\ { 0 } & { ; m _ { t i j h } > \\hat { m } _ { t j h } ^ { l _ { I } } } \\end{array} \\right.\n$$",
+        "text_format": "latex",
+        "bbox": [
+            333,
+            522,
+            663,
+            566
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "160 The final program memory attention is computed as ",
+        "bbox": [
+            142,
+            568,
+            516,
+            583
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/7f5851a3247de278761e154705908476d92d30012c85ff80f22df974bfe7e41f.jpg",
+        "text": "$$\nw _ { t i j h } = \\mathrm { s i g m o i d } \\left( g _ { t i j h } \\right) c _ { t i j h } + \\left( 1 - \\mathrm { s i g m o i d } \\left( g _ { t i j h } \\right) \\right) l _ { t i j h }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            303,
+            602,
+            694,
+            619
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "161 Since the usage record are computed along the memory accesses, the multi-step Neurocoder utilises   \n162 this attention mechanism better than the single-step Neurocoder, creating different attention styles   \n163 (see Sec. 3.2). The composition the active program $\\mathbf { P } _ { t }$ is illustrated in Appendix’s Fig. 5. ",
+        "bbox": [
+            142,
+            622,
+            825,
+            665
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "2.3 Program Integration via the integration network ",
+        "text_level": 1,
+        "bbox": [
+            163,
+            679,
+            552,
+            695
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "165 Since the working program $\\mathbf { P } _ { t }$ only contains top $r _ { m }$ principal components, it is low-rank and may   \n166 be not flexible enough for sophisticated computation. We propose to enhance $\\mathbf { P } _ { t }$ with a residual   \n167 program $\\mathbf { R } -$ a traditional connection weight trained as the integration network’s parameters, which   \n168 is constant after training w.r.t $t$ . The residual program represents the sum of the remaining less   \n169 important components. To this end, we suppress $\\mathbf { R }$ with a multiplier that is smaller than $\\sigma _ { t r _ { m } } .$ – the   \n170 smallest singular value of the main components - resulting in the integration formula ",
+        "bbox": [
+            140,
+            704,
+            825,
+            790
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "equation",
+        "img_path": "images/5a1ea2eaac36e14e3d2dca9766e69d0bd941880cfbe050d8a72f9f696febc4cf.jpg",
+        "text": "$$\nW _ { t } = \\mathbf { P } _ { t } + w _ { t } ^ { r } \\sigma _ { t r _ { m } } \\mathbf { R }\n$$",
+        "text_format": "latex",
+        "bbox": [
+            423,
+            808,
+            575,
+            825
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "171 where $w _ { t } ^ { r } = \\mathrm { s i g m o i d } \\left( f _ { \\phi } \\left( x _ { t } \\right) \\right)$ is an adaptive gating value that controls the contribution of the   \n172 residual program. $f _ { \\phi }$ is the integration network in the Program Controller and hence, in our imple  \n173 mentation, the integration control signal sent by the Program Controller is $\\lambda _ { t } ^ { p } = \\{ w _ { t } ^ { r } , \\sigma _ { t r _ { m } } \\}$ . We   \n174 note that in our experiments, the program integration can be disabled ( $W _ { t }$ is directly set to $\\mathbf { P } _ { t }$ ) to   \n175 prove the contribution of $\\mathbf { P } _ { t }$ or reduce the number of parameters. The working program $W _ { t }$ is then   \n176 used by the Main Network to execute the input data $x _ { t }$ (see (Fig. 1 (a))). For example, with linear   \n177 classifier Main Network, the execution is $y _ { t } = x _ { t } W _ { t }$ . Appendix’s Table 2 summarises the notations   \n178 used for important parameters of Neurocoder. ",
+        "bbox": [
+            140,
+            827,
+            825,
+            912
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/f992f503335ad0a3f753e1c8f03e87de2fd352b9c040fd798ec6c09805c37c77.jpg",
+        "image_caption": [
+            "Figure 2: (a) MNIST test set classification error vs the number of steps $( J )$ in Neurocoder (blue), compared with a linear classifier (red). (b) 1st column: Digit images; Middle column: Single-step attention weights for 30 slots in $\\mathbf { M } _ { U }$ (vertical axis) for first 3 singular vectors (horizontal axis) for each digit; Last column: Multi-step attention weights for 10 slots in $\\mathbf { M } _ { U }$ (vertical axis) for first 3 singular vectors (horizontal axis). Multi-step attention is able to produce far more diverse patterns with fewer slots - 10 slots compared to single-step 30 slots. (c) Two attention patterns of singlestep Neurocoder. The binary decision tree derived from single-step Neurocoder’s attention patterns. The two patterns across components represent the decisions going up and down across the binary tree. Visualisation for (d) multi-step $J = 5$ , 20 memory slots) and (e) single-step $J = 1$ , 10 memory slots) cases showing while processing a sequence of the polynomial auto-regression task. The Neurocoder’s attentions to $\\mathbf { M } _ { U }$ that form the first component of the active program are shown over sequence timesteps (upper) with Neurocoder’s $y _ { t }$ prediction (orange) and ground truth (blue) (lower). The vertical dash green lines separate polynomial chunks. Each chuck represents a local pattern, and thus ideally requires a specific active program to compute the input $x _ { t }$ . Although both predict well, only the multi-step Neurocoder discovers the chunk boundaries, assigning program attention to the first component in accordance with sequence changes. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
+            87,
+            826,
+            337
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            143,
+            618,
+            823,
+            646
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "179 3 Results ",
+        "text_level": 1,
+        "bbox": [
+            142,
+            691,
+            266,
+            708
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "180 To demonstrate the flexibility of Neurocoder framework, we consider different learning paradigms:   \n181 instance-based, sequential, multi-task and continual learning. We do not focus on breaking perfor  \n182 mance records by augmenting state-of-the-art models with Neurocoder. Rather our inquiry is on   \n183 re-coding feed-forward layers with the Neurocoder’s programs and testing on varied data types to   \n184 demonstrate its intrinsic properties. For some experiments, we include ablation studies.   \n185 We compare the performance of diverse Main Networks (MN) with and without Neurocoder. We   \n186 also augment the Main Networks with other recent conditional computing methods, either modular   \n187 (sparse Mixture of Experts, Neural Stored-program Memory) or monolithic (HyperNets, FiLM) to   \n188 form stronger baselines across our experiments. In our experiments, we always apply Neurocoder   \n189 to all layers of multi-layer perceptrons (MLP) or just the final feed-forward layer of deep CNN   \n190 networks (LeNet, DenseNet, ResNet), RNNs (GRU, LSTM), MANN (NTM). Other competitors   \n191 such as MOE, NSM, HyperNet and FiLM are applied to the Main Networks in the same manner. ",
+        "bbox": [
+            142,
+            738,
+            823,
+            808
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            142,
+            814,
+            825,
+            911
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/7e25b32d05c9ae7db2df1ea7bbd96b41284179be7b8a5c34b278f599cab1c811.jpg",
+        "image_caption": [
+            "Figure 3: Learning curves (mean and std. over 5 runs) on representative Atari 2600 games. All baselines are applied to the actor/critic networks in the A3C agent. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            173,
+            88,
+            825,
+            236
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "192 3.1 Instance-based learning - Object Recognition ",
+        "text_level": 1,
+        "bbox": [
+            142,
+            300,
+            526,
+            315
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We tested Neurocoder on instance-based learning through classical image classification tasks using MNIST [24] and CIFAR [21] datasets. The first experiment interpreted Neurocoder’s behaviour in classifying digits into 10 classes $( 0 - 9 )$ using linear classifier Main Network. With equivalent model size, Neurocoder using the novel recurrent attention surpasses the performance of the linear classifier [24] by up to $5 \\%$ (Fig. 2 (a)). ",
+        "bbox": [
+            165,
+            327,
+            823,
+            396
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "To differentiate the input, Neurocoder attends to different components of the active program to guide the decision-making process. Fig. 2 (b) shows single-step and multi-step attention to the first 3 singular vectors for each digit across memory slots. Multi-step attention produces richer patterns compared to single-step Neurocoder that manages only 2 attention weight patterns. ",
+        "bbox": [
+            165,
+            402,
+            823,
+            458
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Fig. 2 (c) illustrates how Neurocoder performs modular learning by showing the attention assignment for top 3 singular vectors as a binary decision tree. Digits under the same parental node share similar attention paths, and thereby similar active programs. Some digits look unique (e.g. 7) resulting in active programs composed of unique attention paths, discriminating themselves early in the decision tree. Some digits (e.g. 0 and 9) share the same attention pattern for the first 3 components and are thus unclassifiable. They can only be distinguished by considering more singular vectors. ",
+        "bbox": [
+            173,
+            464,
+            823,
+            547
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "We integrated Neurocoder with deep networks - 5-layer LeNet and 100-layer DenseNet - and tested on CIFAR datasets. Neurocoder significantly outperformed the original Main Networks with performance gain $1 - 5 \\%$ . Compared with recent conditional computing models such as sparse Mixture of Experts (MOE [35]) and Neural Stored-program Memory (NSM [22]), Neurocoder required a tenth of the number of parameters and performed better by up to $8 - 1 0 \\%$ (see Appendix’s Table 3). ",
+        "bbox": [
+            165,
+            554,
+            825,
+            623
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "3.2 Sequential learning - Adaption to sequence changes and game playing using reinforcement learning ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            641,
+            738,
+            670
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Recurrent neural networks (RNN) can learn from sequential data by updating the hidden states of the networks. However, this does not suffice when local patterns shift, as is often the case. We now demonstrate that Neurocoder helps RNNs overcome this limitation by composing diverse programs to handle sequence changes. ",
+        "bbox": [
+            168,
+            680,
+            825,
+            736
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "Synthetic polynomial auto-regression We created a simple auto-regression task in which data points are sampled from polynomial function chunks that change over time. The Main Network is a strong RNN–Gated Recurrent Unit (GRU [5]). We found that GRU integrated with a single-step or multi-step Neurocoder converged much faster than all other baselines. The other conditional computing counterparts (HyperNet [13], FiLM [28]) adapt by re-scaling weights or activation of the GRU, which were shown inferior to our modular approach (Appendix’s Fig. 6). ",
+        "bbox": [
+            171,
+            752,
+            823,
+            835
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "225 Visualising the first singular vector attention weights in $\\mathbf { M } _ { U }$ , we find that the multi-step attention   \n226 Neurocoder changes its attention following polynomial changes - it attends to the same singular pro  \n227 gram when processing data from the same polynomial and alters attention for data from a different   \n228 polynomial (Fig. 2(d)). In contrast, the single-step Neurocoder only changes its attention when   \n229 there is a remarkable change in $y$ -coordinate values (Fig. 2(e)). Although single-step Neurocoder   \n230 converges well, it did not discover the underlying structure of the data, and thus underperformed   \n231 the multi-step Neurocoder. We hypothesise that when recurrence is employed, usage-based atten  \n232 tion takes effect, stipulating better memory utilisation and diverse attentions over timesteps. We ran   \n233 multi-step Neurocoder without usage-based attention. The results were worse than the full multi  \n234 step Neurocoder, which confirms our hypothesis (Appendix’s Fig. 6). ",
+        "bbox": [
+            142,
+            842,
+            823,
+            911
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "table",
+        "img_path": "images/06ee98d5e15fb67549b39c75f337a7184d9d269eefcccfe677f8248a1273b0b6.jpg",
+        "table_caption": [],
+        "table_footnote": [
+            "Table 1: Incremental domain continual learning with Split MNIST. Final test accuracy (mean and std.) over 10 runs. "
+        ],
+        "table_body": "<table><tr><td>Method</td><td>MN (MLP[17])</td><td>MN (MLP ours)</td><td>NSM</td><td>Neurocoder</td></tr><tr><td>Adam</td><td>55.16±1.38</td><td>53.55±1.27</td><td>54.85±2.81</td><td>58.46±0.46</td></tr><tr><td>Adagrad</td><td>58.08±1.06</td><td>57.83±2.74</td><td>58.42±1.87</td><td>62.28±4.03</td></tr><tr><td>L2</td><td>66.00±3.73</td><td>64.37±2.40</td><td>62.83±7.21</td><td>69.89±1.72</td></tr><tr><td>SI</td><td>64.76±3.09</td><td>64.41±3.36</td><td>64.36±2.99</td><td>67.96±3.22</td></tr><tr><td>EWC</td><td>58.85±2.59</td><td>58.41±2.37</td><td>58.12±3.24</td><td>65.66±1.25</td></tr><tr><td>O-EWC</td><td>57.33±1.44</td><td>57.78±1.84</td><td>58.55±3.40</td><td>73.97±1.50</td></tr></table>",
+        "bbox": [
+            236,
+            88,
+            761,
+            189
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            142,
+            256,
+            823,
+            327
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Atari game reinforcement learning We used reinforcement learning as a further testbed to show the ability to adapt to environmental changes. We performed experiments on several Atari 2600 games [3] wherein the agent was implemented as the Asynchronous Advantage Actor-Critic (A3C [26]). In the Atari platform, agents are allowed to observe the screen snapshot of the games and act to earn the highest score. We augmented the A3C by employing Neurocoder’s working programs for feed-forward layers of the actor and critic networks, aiming to decompose the policy and value function into singular programs that were selected depending on the game state. ",
+        "bbox": [
+            169,
+            343,
+            825,
+            440
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "242 Frostbite and Montezuma’s Revenge. These games are known to be challenging for A3C and other   \n243 algorithms [26]. We trained A3C and HyperNet-based A3C for over 300 million steps, yet these   \n244 models did not show any sign of learning, performing equivalently to random agents. For such com  \n245 plicated environments with sparse rewards, both the monolithic neural networks and the HyperNet’s   \n246 unstored fast-weights fail to learn (almost zero scores). In contrast, Neurocoder enabled A3C to   \n247 achieve from 1, 500 to 3, 000 scores on these environments (Fig. 3), confirming the importance of   \n248 decomposing a complex solution to smaller, simple stored programs. ",
+        "bbox": [
+            142,
+            446,
+            825,
+            545
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "49 3.3 Multi-task learning - Solving mutliple algorithms simultenously ",
+        "text_level": 1,
+        "bbox": [
+            156,
+            563,
+            653,
+            578
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Here we explore the modular learning capability of Neurocoder in multi-task setting. Inspired by algorithmic sequencing tasks [22], we created a challenging sequential multi-task benchmark wherein the input sequence is a series of sub-sequences from 4 algorithms: Copy, Repeat Copy, Associative Recall and Priority Sort [11]. Each sub-sequence, following a task identification vector, represents the input for each task. In each input sequence, $n$ tasks were sampled from the set of 4 algorithms randomly with replacement and the output sequences were created correspondingly. ",
+        "bbox": [
+            171,
+            589,
+            823,
+            671
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We trained a MANN–Neural Turing Machine (NTM [11]) Main Network with FiLM, HyperNet and our Neurocoder augmentation on sequences of $n = 4$ tasks, and tested with sequences of $n = 4$ and $n = 8$ tasks. Appendix’s Fig. 7 demonstrates that Neurocoder was performant in both test settings, not only achieving lowest error on $n = 4$ , but also being the only one generalised well to $n = 8$ scenario, which was unseen during training. ",
+        "bbox": [
+            171,
+            679,
+            825,
+            748
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "3.4 Continual learning $-$ Learning tasks sequentially without catastrophic forgetting ",
+        "text_level": 1,
+        "bbox": [
+            161,
+            767,
+            767,
+            782
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In continual learning, standard neural networks often suffer from “catastrophic forgetting” in which they cannot retain knowledge acquired from old tasks upon learning new ones [10]. Our Neurocoder offers natural mitigation of such catastrophic forgetting in neural networks by attending to different singular programs whilst learning different tasks. ",
+        "bbox": [
+            174,
+            794,
+            823,
+            849
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "In this case, in addition to the Main Network, we examine several continual learning algorithms with and without Neurocoder. These algorithms, including Elastic Weight Consolidation (EWC [41]) and Synaptic Intelligence (SI [41]), work by regularising the loss function and thus can be easily combined with Neurocoder by modifying the loss $\\mathcal { L } _ { t a s k }$ . We demonstrate that Neurocoder ",
+        "bbox": [
+            169,
+            856,
+            823,
+            911
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "70 can improve these continual learning algorithms without requiring additional assumptions as in other   \n1 approaches [25, 36, 34] that either utilise task embedding or replay memory. ",
+        "bbox": [
+            153,
+            90,
+            825,
+            119
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Split MNIST We first considered the split MNIST dataset–a standard continual learning benchmark wherein the original MNIST was split into a 5 2-way classification tasks, consecutively presented to a Multi-layer Perceptron Main Network (MLP). We followed the benchmarking as in [17] in which various optimisers and state-of-the-art continual learning methods were examined under incremental task and domain scenarios. We measured the performance of the MLP versus Neurocoder and NSM under each continual learning method. In both scenarios, Neurocoder was compatible with all continual leaning methods, demonstrating superior performance over MLP and NSM with performance gain between 1 to $1 6 \\%$ (see Appendix’s Table 5 and 1). ",
+        "bbox": [
+            163,
+            136,
+            823,
+            247
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "280 Split CIFAR We verified the scalability of Neurocoder to more challenging datasets. We split   \n281 CIFAR datasets as in the split MNIST, resulting in 5-task 2-way split CIFAR10 and a 20-task 5-way   \n282 split CIFAR100. We used Main Network ResNet [15]–a very deep CNN architecture.   \n283 When we stressed the orthogonal loss $a = 1 0$ ) and used bigger program memory (100 slots), Neu  \n284 rocoder improved ResNet classification by $1 5 \\%$ and $1 0 \\%$ on CIFAR10 and CIFAR100, respectively.   \n285 When we integrated Neurocoder with Synaptic Intelligence (SI [41]), the performance was further   \n286 improved, maintaining a stable performance above $8 0 \\%$ accuracy for CIFAR10 and outperforming   \n287 using SI alone by $1 0 \\%$ for CIFAR100 (see Appendix’s Fig. 8). ",
+        "bbox": [
+            143,
+            263,
+            823,
+            306
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            142,
+            311,
+            823,
+            382
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "288 4 Discussion ",
+        "text_level": 1,
+        "bbox": [
+            143,
+            402,
+            294,
+            420
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Our experiments demonstrate that Neurocoder is capable of re-coding Neural Programs in distinctive neural networks, amplifying their capabilities in diverse learning scenarios: instance-based, sequential, multi-task and continual learning. This consistently results in significant performance increase, and further creates novel robustness to pattern shift and catastrophic forgetting. This ability for each architecture to re-code itself is made possible without changing the way it is trained, or majorly increasing the number of parameters it needs to learn (see Appendix Table 7). ",
+        "bbox": [
+            171,
+            435,
+            823,
+            518
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "The MNIST problem illustrates the reasoning process of Neurocoder when classifying digit images wherein its singular program assignment resembles a binary tree decision-making process - it shows how some singular programs are shared, others are not. The polynomial auto-regression problem highlights the importance of efficient memory utilisation in re-constructing the working program enabling discovery of hidden structures in sequential data. Training our framework with reinforcement learning, we enable neural agents to solve complex games wherein traditional methods fail or learn slowly. Neurocoder also works well with multi-task setting, as shown in the challenging multi-algorithm benchmark. Finally, continual learning problems show that Neurocoder mitigates catastrophic forgetting efficiently under different learning settings/algorithms. ",
+        "bbox": [
+            165,
+            525,
+            825,
+            650
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "304 Our solution offers a single framework that is scalable and adaptable to various problems and learn  \n305 ing paradigms. Unlike previous attempts to employ a bank of separate big programs [20, 35, 22],   \n306 Neurocoder maintains only shareable, smaller components that can reconstruct the whole program   \n307 space, thereby heavily utilising the parameters and preventing the model from proliferating. We   \n308 note that Neurocoder is orthogonal to approaches employing tensor decomposition to reduce the   \n309 number of parameters or hasten the computation [27, 23]. Neurocoder composes rather than decom  \n310 pose the neural weights. Our aim is not only to enable efficient parameter usage, but also achieve   \n311 general-purpose computing power, outperforming other methods in numerous learning problems.   \n312 One limitation of this work is the number of additional hyperparameters, which prevents us from   \n313 fully tuning Neurocoder. Our research aims to add new capabilities to current neural networks to   \n314 improve their performance and make them robust in different learning scenario. Hence, we do   \n315 not see any intermediate negative societal impact. In future work, we will extend Neurocoder’s   \n316 application beyond feed-forward layers. It would be interesting to efficiently replace all neural layers   \n317 including CNN or Transformer by Neurocoder’s programs. We can also further extend Neurocoder’s   \n318 ability by allowing a growing Program Memory, in which the model decides to add or erase memory   \n319 slots as the number of data patterns grows or shrinks beyond the current program space’s capacity.   \n320 Such a system represents a more flexible general-purpose computer that can dynamically allocate   \n321 computing resources by itself without human pre-specification. ",
+        "bbox": [
+            140,
+            656,
+            825,
+            767
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            140,
+            772,
+            825,
+            911
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "References   \n[1] Jacob Andreas, Marcus Rohrbach, Trevor Darrell, and Dan Klein. Neural module networks. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 39–48, 2016.   \n[2] Dzmitry Bahdanau, Kyunghyun Cho, and Yoshua Bengio. Neural machine translation by jointly learning to align and translate. In International Conference on Learning Representations, 2015.   \n[3] Marc G Bellemare, Yavar Naddaf, Joel Veness, and Michael Bowling. The arcade learning environment: An evaluation platform for general agents. Journal of Artificial Intelligence Research, 47:253–279, 2013.   \n[4] Yoshua Bengio, Nicholas Léonard, and Aaron Courville. Estimating or propagating gradients through stochastic neurons for conditional computation. arXiv preprint arXiv:1308.3432, 2013.   \n[5] Kyunghyun Cho, Bart van Merriënboer, Caglar Gulcehre, Dzmitry Bahdanau, Fethi Bougares, Holger Schwenk, and Yoshua Bengio. Learning phrase representations using RNN encoder– decoder for statistical machine translation. In Conference on Empirical Methods in Natural Language Processing (EMNLP), pages 1724–1734. Association for Computational Linguistics, October 2014.   \n[6] JC Eccles. The modular operation of the cerebral neocortex considered as the material basis of mental events. Neuroscience, 6(10):1839–1855, 1981.   \n[7] Gerald M Edelman. Neural darwinism: selection and reentrant signaling in higher brain function. Neuron, 10(2):115–125, 1993.   \n[8] Gerald M Edelman and Vernon B Mountcastle. The mindful brain: cortical organization and the group-selective theory of higher brain function. Massachusetts Inst of Technology Pr, 1978.   \n[9] Richard SJ Frackowiak. Human brain function. Elsevier, 2004.   \n[10] Robert M French. Catastrophic forgetting in connectionist networks. Trends in cognitive sciences, 3(4):128–135, 1999.   \n[11] Alex Graves, Greg Wayne, and Ivo Danihelka. Neural turing machines. arXiv preprint arXiv:1410.5401, 2014.   \n[12] Alex Graves, Greg Wayne, Malcolm Reynolds, Tim Harley, Ivo Danihelka, Agnieszka Grabska-Barwinska, Sergio Gómez Colmenarejo, Edward Grefenstette, Tiago Ramalho, John ´ Agapiou, et al. Hybrid computing using a neural network with dynamic external memory. Nature, 538(7626):471–476, 2016.   \n[13] David Ha, Andrew M. Dai, and Quoc V. Le. Hypernetworks. In International Conference on Learning Representations, 2017.   \n[14] Bart LM Happel and Jacob MJ Murre. Design and evolution of modular neural network architectures. Neural networks, 7(6-7):985–1004, 1994.   \n[15] Kaiming He, Xiangyu Zhang, Shaoqing Ren, and Jian Sun. Deep residual learning for image recognition. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 770–778, 2016.   \n[16] Sepp Hochreiter and Jürgen Schmidhuber. Long short-term memory. Neural computation, 9(8):1735–1780, 1997.   \n[17] Yen-Chang Hsu, Yen-Cheng Liu, Anita Ramasamy, and Zsolt Kira. Re-evaluating continual learning scenarios: A categorization and case for strong baselines. In NeurIPS Continual learning Workshop, 2018.   \n[18] Gao Huang, Zhuang Liu, Laurens Van Der Maaten, and Kilian Q Weinberger. Densely connected convolutional networks. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 4700–4708, 2017.   \n[19] D.H. Hubel. Eye, Brain, and Vision. Scientific American Library series. Scientific American Library, 1988.   \n[20] Robert A Jacobs, Michael I Jordan, Steven J Nowlan, and Geoffrey E Hinton. Adaptive mixtures of local experts. Neural computation, 3(1):79–87, 1991.   \n[21] Alex Krizhevsky, Geoffrey Hinton, et al. Learning multiple layers of features from tiny images. TR-2009, 2009.   \n[22] Hung Le, Truyen Tran, and Svetha Venkatesh. Neural stored-program memory. In International Conference on Learning Representations, 2020.   \n[23] V Lebedev, Y Ganin, M Rakhuba, I Oseledets, and V Lempitsky. Speeding-up convolutional neural networks using fine-tuned cp-decomposition. In International Conference on Learning Representations, 2015.   \n[24] Yann LeCun, Léon Bottou, Yoshua Bengio, and Patrick Haffner. Gradient-based learning applied to document recognition. Proceedings of the IEEE, 86(11):2278–2324, 1998.   \n[25] David Lopez-Paz and Marc’Aurelio Ranzato. Gradient episodic memory for continual learning. In Advances in neural information processing systems, pages 6467–6476, 2017.   \n[26] Volodymyr Mnih, Adria Puigdomenech Badia, Mehdi Mirza, Alex Graves, Timothy Lillicrap, Tim Harley, David Silver, and Koray Kavukcuoglu. Asynchronous methods for deep reinforcement learning. In International conference on machine learning, pages 1928–1937, 2016.   \n[27] Alexander Novikov, Dmitrii Podoprikhin, Anton Osokin, and Dmitry P Vetrov. Tensorizing neural networks. In Advances in neural information processing systems, pages 442–450, 2015.   \n[28] Ethan Perez, Florian Strub, Harm De Vries, Vincent Dumoulin, and Aaron Courville. FiLM: Visual Reasoning with a General Conditioning Layer. In AAAI Conference on Artificial Intelligence, New Orleans, United States, February 2018.   \n[29] Clemens Rosenbaum, Tim Klinger, and Matthew Riemer. Routing networks: Adaptive selection of non-linear functions for multi-task learning. In International Conference on Learning Representations, 2018.   \n[30] David E Rumelhart, Geoffrey E Hinton, and Ronald J Williams. Learning representations by back-propagating errors. Nature, 323(6088):533–536, 1986.   \n[31] Adam Santoro, Sergey Bartunov, Matthew Botvinick, Daan Wierstra, and Timothy Lillicrap. Meta-learning with memory-augmented neural networks. In International conference on machine learning, pages 1842–1850, 2016.   \n[32] Jürgen Schmidhuber. Making the world differentiable: On using self-supervised fully recurrent neural networks for dynamic reinforcement learning and planning in non-stationary environm nts. TR FKI-126-90, 1990.   \n[33] Jürgen Schmidhuber. Learning to control fast-weight memories: An alternative to dynamic recurrent networks. Neural Computation, 4(1):131–139, 1992.   \n[34] Joan Serra, Didac Suris, Marius Miron, and Alexandros Karatzoglou. Overcoming catastrophic forgetting with hard attention to the task. In International Conference on Machine Learning, pages 4548–4557, 2018.   \n[35] Noam Shazeer, Azalia Mirhoseini, Krzysztof Maziarz, Andy Davis, Quoc V. Le, Geoffrey E. Hinton, and Jeff Dean. Outrageously large neural networks: The sparsely-gated mixture-ofexperts layer. In International Conference on Learning Representations, 2017.   \n[36] Hanul Shin, Jung Kwon Lee, Jaehong Kim, and Jiwon Kim. Continual learning with deep generative replay. In Advances in Neural Information Processing Systems, pages 2990–2999, 2017.   \n[37] A.M Turing. On computable numbers, with an application to the entscheidungsproblem. In Proceedings of the London Mathematical Society, 1936.   \n[38] Ashish Vaswani, Noam Shazeer, Niki Parmar, Jakob Uszkoreit, Llion Jones, Aidan N Gomez, Łukasz Kaiser, and Illia Polosukhin. Attention is all you need. In Advances in neural information processing systems, pages 5998–6008, 2017.   \n[39] Christoph von der Malsburg. The correlation theory of brain function, 1981.   \n[40] John Von Neumann. First draft of a report on the edvac. IEEE Annals of the History of Computing, 15(4):27–75, 1993.   \n[41] Friedemann Zenke, Ben Poole, and Surya Ganguli. Continual learning through synaptic intelligence. Proceedings of machine learning research, 70:3987, 2017. ",
+        "bbox": [
+            151,
+            88,
+            828,
+            915
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            158,
+            47,
+            828,
+            921
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            151,
+            89,
+            828,
+            323
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "1. For all authors... ",
+        "bbox": [
+            214,
+            116,
+            339,
+            130
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]   \n(b) Did you describe the limitations of your work? [Yes] See Discussion and Appendix’s \"Training procedure and hyper-parameter selections.\"   \n(c) Did you discuss any potential negative societal impacts of your work? [Yes] See Discussion.   \n(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] ",
+        "bbox": [
+            238,
+            135,
+            825,
+            253
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "2. If you are including theoretical results... ",
+        "bbox": [
+            214,
+            257,
+            493,
+            272
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "(a) Did you state the full set of assumptions of all theoretical results? [N/A] (b) Did you include complete proofs of all theoretical results? [N/A] ",
+        "bbox": [
+            235,
+            276,
+            736,
+            308
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "3. If you ran experiments... ",
+        "bbox": [
+            212,
+            311,
+            393,
+            325
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [No] Data is public, provided with link. Code will be avaialble after published. All training details are available and can be used to implement and reproduce the results.   \n(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] See Appendix’s \"Training procedure and hyper-parameter selections.\"   \n(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [Yes] See Experimental Results.   \n(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] See Appendix’s \"Training procedure and hyper-parameter selections. ",
+        "bbox": [
+            238,
+            330,
+            826,
+            489
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets... ",
+        "bbox": [
+            215,
+            494,
+            823,
+            510
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "(a) If your work uses existing assets, did you cite the creators? [Yes]   \n(b) Did you mention the license of the assets? [No] All assets are public. We will mention the license detail after the paper is published.   \n(c) Did you include any new assets either in the supplemental material or as a URL? [No]   \n(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [No]   \n(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [No] ",
+        "bbox": [
+            238,
+            513,
+            825,
+            635
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "5. If you used crowdsourcing or conducted research with human subjects... ",
+        "bbox": [
+            214,
+            638,
+            707,
+            654
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]   \n(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]   \n(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A] ",
+        "bbox": [
+            238,
+            657,
+            825,
+            747
+        ],
+        "page_idx": 12
+    }
+]

parse/train/O0ow9RQW1nP/O0ow9RQW1nP_middle.json ADDED Viewed

The diff for this file is too large to render. See raw diff

parse/train/O0ow9RQW1nP/O0ow9RQW1nP_model.json ADDED Viewed

The diff for this file is too large to render. See raw diff

parse/train/ZS394D3djsg/ZS394D3djsg.md ADDED Viewed

	@@ -0,0 +1,457 @@

+# Fitting large mixture models using stochastic component selection
+Anonymous Author(s)
+Affiliation
+Address
+email
+# Abstract
+1 Traditional methods for unsupervised learning of finite mixture models require to
+2 evaluate the likelihood of all components of the mixture. This becomes computa
+3 tionally prohibitive when the number of components is large, as it is, for example,
+4 in the sum-product (transform) networks. As a remedy, we propose an approach
+5 combining the expectation maximization and the Metropolis-Hastings algorithm
+6 to evaluate only a small number of, stochastically sampled, components, thus
+7 substantially reducing the computational cost. We put emphasis on generality of
+8 our method, equipping it with the ability to train both shallow and deep mixture
+9 models which involve complex, and possibly nonlinear, transformations. The
+10 performance of our method is illustrated in a variety of synthetic and real-data
+11 contexts, considering deep models, such as mixtures of normalizing flows and
+12 sum-product (transform) networks.
+# 13 1 Introduction
+14 Finite mixture models [40] constitute a fundamental class of density estimation models. They
+15 have been successfully applied in diverse fields, including bioinformatics [49], econometrics [10],
+16 engineering [33], etc. A mixture model relies on a weighted sum of probability distributions—here
+17 referred to as components—to cluster $N$ unlabelled datapoints into $K$ categories. The traditional
+18 maximum likelihood techniques train the model by optimizing either (i) the marginal likelihood via
+19 gradient-descent [50] or (ii) the evidence lower bound via variational methods [4], including the
+20 expectation-maximization (EM) [13]. The dependence structure among approximate, variational,
+21 distributions then ranges from the fully independent (mean-field) [25] to fully dependent [30]. The
+22 sampling-based techniques target the posterior distribution using sequential Monte Carlo [9] or
+23 Markov chain Monte Carlo [52], e.g. via the Gibbs [34] or Metropolis-Hastings sampling [38]. The
+24 computational cost of these methods typically scales with $\mathcal { O } ( T K N D )$ operations, where $N$ and $K$
+25 are defined above, $T$ is the number of iterations and $D$ is the dimension of data.
+26 Various methods to decrease the computational cost via any factor in $\mathcal { O } ( T K N D )$ have been proposed.
+27 $T$ can be lowered by proper initialization, e.g. the optimal seeding [5]; an efficient step-size schedule,
+28 e.g. the line-search [58]; or increased estimation precision, e.g. the variance reduction [8]. $N$ is often
+29 reduced using the coreset methods, which approximate the original dataset by a weighted dataset
+30 such that the exact and approximate marginal likelihoods are close. The weighted variants of the
+31 variational [17, 59, 6] and sampling-based [39] methods then process the coresets. Reducing $D$ relies
+32 on the compression of data into smaller representations via random projections [53, 2], which is
+33 achieved in two ways: (i) each data item is projected into an individual representation [11]; (ii) all
+34 data items are projected into an overall representation, commonly referred to as sketch [28, 22].
+35 Nevertheless, all the aforementioned techniques—including those with reduced computational cost—
+36 evaluate all $K$ components. This is very demanding for large models, and the problem is even more
+37 severe for mixtures involving intricate models, such as neural networks [21, 42], Gaussian processes
+38 [57], normalizing flows [48]; or deep mixtures, including sum-product (transform) networks [45, 47],
+39 deep Gaussian mixture models [55], etc. In spite of this, a little attention has been paid to the design
+40 of algorithms which does not evaluate all $K$ components. The notable exceptions are the sparse EM
+41 algorithm [24] and the truncated variational EM algorithm [18], see Table 1 and Section 5 for details.
+42 Moreover, the methods are mostly tailored for a specific class of mixture models, e.g. the Gaussian
+43 mixture models.
+Table 1: The computational features of various EM algorithms. We compare whether the methods (i) perform the computations with a reduced number of data (minibatching), (ii) update a lower number of statistics, (iii) make less evaluations of the conditional likelihood, and (iv) are suitable for training of deep models. Here, EM, SA, S, T, MC and MH stand for expectation-maximization, stochastic approximation, sparse, truncated, Monte Carlo and Metropolis-Hastings, respectively.
+<table><tr><td>Feature/Algorithm</td><td>EM [13]</td><td>SAEM [44]</td><td>SSAEM [24]</td><td>TSAEM [18]</td><td>MCSAEM [1]</td><td>MHSAEM (ours)</td></tr><tr><td>B&lt;Ndatapoints</td><td>×</td><td></td><td></td><td>√</td><td></td><td></td></tr><tr><td>M&lt;K statistics</td><td>xx</td><td></td><td></td><td></td><td>厂</td><td></td></tr><tr><td>M&lt;Klikelihoods</td><td></td><td>xx</td><td>×</td><td></td><td>X</td><td></td></tr><tr><td>deep models</td><td>×</td><td>×</td><td>×</td><td>X</td><td>×</td><td></td></tr></table>
+44 In this paper, we make the following contributions:
+45 • We propose an EM-based algorithm which relies on the MH sampler to stochastically evaluate less
+46 components in mixture models, substantially reducing the computational cost.
+7 • We design our method to enable optimization of fairly generic EM objective functions, making it
+48 suitable for training of both shallow and deep mixture models.
+49 • We apply our approach to Gaussian mixture mdoels (GMMs) and their generalizations: sum
+50 product-transform networks (SPTNs) and mixtures of real-valued non-volume preserving (real
+51 NVP) flows [15], reaching approximately $1 0 0 \times$ speed-up compared to state-of-the-art methods.
+# 52 2 Problem formulation
+A finite mixture model characterizes the relation between an observed (known) variable, 53 $\boldsymbol { x } \in \times \subseteq \mathbb { R } ^ { D }$ , 54 and a latent (unknown) variable, $z \in Z : = \{ 1 , \dots , K \}$ , via the marginal (incomplete-data) likelihood 55 in the following form:
+$$
+p _ { \theta } ( x ) = \sum _ { k = 1 } ^ { K } p _ { \eta _ { k } } ( x | z = k ) p _ { \pi _ { k } } ( z = k ) ,
+$$
+56 where $\theta : = ( \pi _ { 1 } , \eta _ { 1 } , \dots , \pi _ { K } , \eta _ { K } ) \in \Theta$ are unknown parameters. Here, $\eta _ { z }$ are the parameters of the 57 conditional likelihood, $p _ { \eta _ { z } } ( x | z )$ , and $\pi _ { z }$ is the weight which parameterizes the prior, $p _ { \pi _ { z } } ( z ) = \pi _ { z }$ , and satisfies 58 $0 \leq \pi _ { k } \leq 1$ for each $k \in { \mathord { \mathbb { Z } } }$ and $\textstyle \sum _ { k = 1 } ^ { K } \pi _ { k } = 1$ .
+59 Given a set of independent and identically distributed data, $\mathbf { x } : = ( x _ { i } ) _ { i = 1 } ^ { N }$ , our goal is to learn the
+60 unknown parameters of the marginal log-likelihood,
+$$
+\mathcal { L } ( \theta ) : = \log p _ { \theta } ( \mathbf { x } ) = \sum _ { i = 1 } ^ { N } \log \sum _ { k = 1 } ^ { K } p _ { \eta _ { k } } ( x _ { i } | z _ { i } = k ) p _ { \pi _ { k } } ( z _ { i } = k ) .
+$$
+61 The marginalization in (2) is tractable for almost all forms of $p _ { \eta _ { z } } ( x | z )$ . Indeed, we consider $p _ { \eta _ { z } } ( x | z )$
+62 to belong to an arbitrary family of $\eta _ { z }$ -differentiable probability distributions. However, we assume that
+63 $K$ is high, making the marginalization in (2) computationally costly, thus rendering the optimization
+64 objective presumably intractable. Therefore, we want to design a computationally efficient algorithm,
+65 requiring only $M < K$ evaluations of $p _ { \eta _ { z } } ( x | z )$ at each iteration.
+67 The maximum likelihood estimation seeks the parameters maximizing the marginal log-likelihood,
+68 $\theta ^ { M L } : = \arg \operatorname* { m a x } _ { \theta \in \Theta } \mathcal { L } ( \theta )$ . The traditional EM algorithm [13] addresses this task indirectly, i.e. by
+69 optimizing the evidence lower bound (ELBO),
+$$
+\mathcal { L } ( \theta ) \geq \mathcal { Q } ( \theta ) + \mathcal { H } ( \hat { \theta } ) : = \mathrm { E L B O } ( \hat { \theta } ) ,
+$$
+where 70 $\mathcal { H } ( \hat { \theta } ) : = - \mathsf E _ { p _ { \hat { \theta } } ( \mathbf { z } | \mathbf { x } ) } [ \log p _ { \hat { \theta } } ( \mathbf { z } | \mathbf { x } ) ]$ is the differential entropy at an estimate, $\hat { \theta } \in \Theta$ , and
+$$
+\mathcal { Q } ( \theta ) : = \mathsf { E } _ { p _ { \hat { \theta } } ( \mathbf { z } | \mathbf { x } ) } [ \log p _ { \theta } ( \mathbf { z } , \mathbf { x } ) ] = \sum _ { i = 1 } ^ { N } \sum _ { k = 1 } ^ { K } p _ { \theta } ( z _ { i } = k | x _ { i } ) \log p _ { \theta } ( z _ { i } = k , x _ { i } )
+$$
+71 is the EM objective function. Here, $p _ { \theta } ( \mathbf { z } , \mathbf { x } )$ is the joint (complete-data) likelihood, and $p _ { \theta } ( \mathbf { z } | \mathbf { x } )$ is
+72 the posterior distribution over the latent variables $\dot { \mathbf { z } } : = ( z _ { i } ) _ { i = 1 } ^ { N }$ . Given an initial value, $\theta _ { 0 }$ , the EM
+73 algorithm produces a sequence of estimates, $( \theta _ { t } ) _ { t = 1 } ^ { T }$ , by alternating between the expectation (E) and
+74 maximization (M) steps,
+$$
+\begin{array} { r l } & { \mathrm { E \mathrm { - } s t e p } ; ~ \mathcal { Q } _ { t - 1 } ( \theta ) , } \\ & { \mathrm { M \mathrm { - } s t e p } ; ~ \theta _ { t } : = \arg \operatorname* { m a x } _ { \theta \in \Theta } \mathcal { Q } _ { t - 1 } ( \theta ) . } \end{array}
+$$
+75 This sequence is guaranteed to monotonically tighten the ELBO, arriving at a local optimum of (2)
+76 under mild regularity assumptions [56].
+77 The EM algorithm is computationally expensive, since (4) evaluates $p _ { \theta } ( z _ { i } , x _ { i } )$ for each $z _ { i } \in \mathbb { Z }$ and
+78 $i \in ( 1 , \ldots , N )$ . This has to be performed for all $t \in ( 1 , \ldots , T )$ in (5). Albeit the marginal factor,
+79 $p _ { \pi _ { z } } ( z )$ , is just the cheap categorical distribution, the conditional factor, $p _ { \eta _ { z } } ( x | z )$ , typically involves
+80 high-dimensional operations (e.g., the inversion of the full $D \times D$ -dimensional covariance matrices
+81 in the GMMs). Moreover, the M-step (6) is also expensive for large $K$ . This holds despite that (6)
+82 can be reduced to closed-form updates of expected sufficient statistics for $p _ { \eta _ { z } } ( x | z )$ belonging to the
+83 exponential family [44] (again, due to high $D$ ). All in all, the computational complexity of the EM
+84 algorithm scales with $\mathcal { O } ( T D N K )$ .
+If (6) cannot be computed under a closed-form solution, one can resort to direct gradient-descent optimization of $\mathcal { Q } ( \boldsymbol { \theta } )$ , where arg max is replaced by one (or more) step(s) of a gradient descent technique. The EM algorithm is then referred to as the generalized EM algorithm [56].
+# 88 4 The generalized MHSAEM algorithm
+89 We design a version of the generalized EM algorithm suitable for scenarios where (4) can represent
+90 deep, discrete, latent variable models, thus being parameterized by possibly complex nonlinear
+91 transformations. We particularly focus on decreasing the the number of operations in the generalized
+92 EM algorithm from $\mathcal { O } ( T D N K )$ to $\mathcal { O } ( T D B M )$ , where $B \ll N$ and $M \ll K$ .
+# 4.1 E-step
+94 We reduce the cost of evaluating the EM objective function (4) by combining the minibatching (as
+95 used many times before) and the Monte Carlo sampling. Namely, the specific application of the latter
+96 to generic mixture models is the key contribution of this paper.
+97 Minibatching. At each iteration, $t$ , we compute the conditional expectation in (4) only for a subset—
+98 here referred to as a minibatch—of the original full dataset, i.e. $( x _ { i } ) _ { i \in I }$ . Here, $I$ is a set of $B \ll N$
+99 indices, $i$ , sampled uniformly without replacement from $( 1 , \ldots , \dot { N } )$ . This substantially decreases the
+100 necessary computations compared to the full sweep over all $N$ datapoints [23].
+101 Monte Carlo sampling. For each $i \in I$ , we want to draw $M \ll K$ random samples from $p _ { \theta } ( z _ { i } | x _ { i } )$ in
+102 order to obtain a Monte Carlo estimate of (4). The straightforward way to do this would be to draw
+103 the samples directly from $p _ { \theta } ( z _ { i } | x _ { i } )$ . However, direct sampling from $p _ { \theta } ( z _ { i } | x _ { i } )$ does not lead to any
+104 substantial decrease in the number of operations. This is caused by the fact that even for a single
+105 sample of $z _ { i }$ , we have to first compute the normalizing factor, $p _ { \theta } ( x _ { i } )$ , to obtaining the posterior,
+106 $p _ { \theta } ( z _ { i } | x _ { i } )$ . This requires $K$ expensive evaluations of $p _ { \theta } ( z _ { i } , x _ { i } )$ , which is precisely what we want to
+107 avoid. Our approach is to resort to the Markov chain Monte Carlo (MCMC), which allows us to
+108 sample from $p _ { \theta } ( z _ { i } | x _ { i } )$ , with the computational complexity decreasing to only a single evaluation of
+109 $p _ { \theta } ( z _ { i } , x _ { i } )$ per a single sample of $z _ { i }$ .
+110 MCMC methods obviate the computation of the normalizing factor in $p _ { \theta } ( z _ { i } | x _ { i } )$ by simulating a
+111 Markov chain, $( z _ { i , t } ) _ { t = 1 } ^ { T }$ , from a transition kernel, $z _ { i , t } \sim P ( z _ { i , t - 1 } , \cdot )$ , which leaves $p _ { \theta } ( z _ { i } | x _ { i } )$ as its
+112 unique stationary (invariant) distribution, starting from an initial value $z _ { i , 0 }$ . The specific form of $P$
+113 determines the structure of an MCMC method. We chose the Metropolis-Hastings (MH) sampler,
+114 which represents $P ( z _ { i , t - 1 } , z _ { i , t } )$ as follows: given $\bar { z } _ { i } : = z _ { i , t - 1 }$ , draw a sample from the proposal
+115 distribution $z _ { i } \sim q ( \cdot | \bar { z } _ { i } )$ , compute the acceptance ratio,
+$$
+\alpha ( \bar { z } _ { i } , z _ { i } ) : = \operatorname* { m i n } \biggr \{ 1 , \frac { p _ { \eta _ { z _ { i } , t - 1 } } ( x _ { i } | z _ { i } ) \pi _ { z _ { i } , t - 1 } q ( \bar { z } _ { i } | z _ { i } ) } { p _ { \eta _ { \bar { z } _ { i } , t - 1 } } ( x _ { i } | \bar { z } _ { i } ) \pi _ { \bar { z } _ { i } , t - 1 } q ( z _ { i } | \bar { z } _ { i } ) } \biggr \} ,
+$$
+116 and, if $u < \alpha \big ( \bar { z } _ { i } , z _ { i } \big )$ —where $u$ is drawn from a uniform distribution, Uniform $( 0 , 1 )$ —accept the
+117 sample and set $z _ { i , t } = z _ { i }$ ; otherwise, set $z _ { i , t } = \bar { z } _ { i }$ . For each $i \in I$ and $t \in ( 1 , \ldots , T )$ , we repeat
+118 this process $M$ times, construing a set $\mathbf { z } _ { i , t } = ( z _ { i , t } ^ { 1 } , \dots , z _ { i , t } ^ { M } )$ . Therefore, at every current iteration,
+119 120 $t$ , we caking $\bar { z } _ { i } = z _ { i , t - 1 } ^ { M }$ extend the chain from the point where we left at the previous iteration, . Under mild regularity assumptions [52], the chain passes the transiti $t - 1$ , byriod
+121 (the burn-in phase), and the samples can then be used to approximate the conditional expectation in
+122 (4) as follows:
+$$
+\widehat { \mathcal { Q } } _ { t - 1 } ( \theta ) = \frac { 1 } { M } \sum _ { i \in I } \sum _ { z \in \mathbf { z } _ { i , t } } \log p _ { \eta _ { z } } ( x _ { i } | z ) \pi _ { z } .
+$$
+123 Note that, to ensure this approach is truly efficient, we have to draw only $M \ll K$ samples at each
+124 iteration, $t$ ; otherwise, for $M \approx K$ , we may rather compute the exact marginalization in (4), since it
+125 is tractable (but computationally costly).
+# 4.2 M-step
+127 Assume for a moment that (6) with $\mathcal { Q } _ { t - 1 } ( \theta )$ given by (8) has a closed-form solution, yielding an
+128 estimate of $\theta$ . Such an estimate would have a high variance, converging only for $M \to \infty$ and
+129 $T \to \infty$ [19]. The main reason is that the samples would not be reused over the iterations, $t$ ,
+130 thus wasting computational resources. We consider that there is no closed-form solution of (6),
+131 and—to ensure that the samples (and thus computations) are recycled over the iterations—we use
+132 the stochastic approximation (SA) [51] to optimize (8). This is analogous to applying a stochastic
+133 gradient-descent method, $\theta _ { t } = \theta _ { t - 1 } + \gamma _ { t } \nabla _ { \theta } \tilde { \mathcal { Q } } _ { t - 1 } ( \theta )$ , where $\gamma _ { t }$ is the step-size, satisfying the Robbins
+134 Monro constraints, $\begin{array} { r } { \gamma _ { t } \in [ 0 , 1 ] , \sum _ { t \geq 1 } \gamma _ { t } = \infty , \sum _ { t \geq 1 } \gamma _ { t } ^ { 2 } < \infty , } \end{array}$ and $\nabla _ { \theta }$ is the gradient w.r.t. $\theta$ . In this
+135 way, the computations made in $\nabla _ { \boldsymbol { \theta } } \widehat { \mathcal { Q } }$ are accumulated via $\theta _ { t }$ and reused over the iterations.
+136 The parameters $\eta _ { z }$ have a different form based on a specific case of $p _ { \eta _ { z } } ( x | z )$ , whereas $\pi _ { z }$ is a
+137 permanent structure in (1). Therefore, without loss of generality, we split (6) into a generic part and a
+138 fixed part as follows:
+$$
+\begin{array} { r l } & { \eta _ { k , t } = \eta _ { k , t - 1 } + \gamma _ { t } \nabla _ { \eta _ { k } } \widehat { \mathcal { Q } } _ { t - 1 } ( \theta ) , } \\ & { \nu _ { k , t } = \nu _ { k , t - 1 } + \gamma _ { t } \nabla _ { \nu _ { k } } \widehat { \mathcal { Q } } _ { t - 1 } ( \theta ) , } \end{array}
+$$
+139 where—to ensure that the probabilities, $( \pi _ { k , t } ) _ { k = 1 } ^ { K }$ , satisfy the constraints (Section 2)—we transform
+140 $\nabla _ { \pi _ { k } } \widehat { \mathcal { Q } }$ via $\nu _ { k } = \log \pi _ { k }$ and optimize w.r.t. $\nu _ { k }$ . Then, to obtain $( \pi _ { k , t } ) _ { k = 1 } ^ { K }$ from $\nu _ { t } : = ( \nu _ { k , t } ) _ { k = 1 } ^ { K }$ , we
+141 k b  use the softmax function, i.e. $\pi _ { k , t } : = \mathrm { s o f t m a x } ( \pmb { \nu } _ { t } ) _ { k } : = \exp ( \nu _ { k , t } ) / \sum _ { l = 1 } ^ { K } \exp ( \nu _ { l , t } )$ .
+Computing the gradients for all pairs of $( \nu _ { k } , \eta _ { k } ) _ { k = 1 } ^ { K }$ would be inefficient, especially since ${ \bf z } _ { i , t }$ contains only a small number of unique values of Z for $M \ll K$ . Consequently, we compute $\dot { \nabla } _ { \eta _ { k } } \widehat { \mathcal { Q } }$ and $\nabla _ { \nu _ { k } } \widehat { \mathcal { Q } }$ only for $k \in { \mathrm { u n i q u e } } ( \mathbf { z } _ { i , t } )$ . We summarize the proposed approach in Algorithm 1.
+# 4.3 Proposal distribution
+146 The choice of the proposal distribution has a significant impact on the speed of convergence and the computational cost of the proposed algorithm. Here, we discuss various possible choices of 147 $q \big ( z _ { i } | \bar { z } _ { i } \big )$ .
+Input: $\theta _ { 0 }$ , $( \mathbf { z } _ { i , 0 } ) _ { i = 1 } ^ { N }$ , $( \mathbf { x } _ { i } ) _ { i = 1 } ^ { N }$
+Output: $( \theta _ { t } ) _ { t = 1 } ^ { T }$ for $t \in ( 1 , \ldots , T )$ or until convergence do form the set $\dot { I } = ( i _ { j } ) _ { j = 1 } ^ { B }$ by sampling (without replacement) $B$ indices $i \sim ( 1 , \dots , N )$ for $i \in I$ do set $\bar { z } _ { i }$ as the last element of $\mathbf { z } _ { i , t - 1 }$ for $j \in ( 1 , \ldots , M )$ do sample $z _ { i } \sim q ( z _ { i } | \bar { z } _ { i } )$ sample $u \sim \mathrm { U n i f o r m } ( 0 , 1 )$ compute $\alpha ( \bar { z } _ { i } , z _ { i } )$ in (7) if $u < \alpha \big ( \bar { z } _ { i } , z _ { i } \big )$ then set $z _ { i , t } ^ { j } = z _ { i }$ and $\bar { z } _ { i } = z _ { i }$ else set $z _ { i , t } ^ { j } = \bar { z } _ { i }$ end if end for set $\mathbf { z } _ { i , t } = ( z _ { i , t } ^ { 1 } , \dots , z _ { i , t } ^ { M } )$ end for compute (8) compute (9) for $k \in { \mathrm { u n i q u e } } ( \mathbf { z } _ { i , t } )$ compute $\pi _ { k , t } : = \mathrm { s o f t m a x } ( \pmb { \nu } _ { t } ) _ { k }$ for $k \in { \mathord { \mathbb { Z } } }$ end for
+148 Optimal proposal $( O )$ . The optimal proposal distribution is $q ( z _ { i } | \bar { z } _ { i } ) : = q ( z _ { i } ) : = p _ { \theta } ( z _ { i } | x _ { i } )$ . This
+149 ensures that the acceptance rate (7) is always $\alpha ( \bar { z } _ { i } , z _ { i } ) = 1$ . However, the need to perform $K$
+150 expensive evaluations of $p _ { \theta } ( z _ { i } , x _ { i } )$ before sampling from $p _ { \theta } ( z _ { i } | x _ { i } )$ is the reason we resorted to
+151 the MH sampler in the first place. We consider this case only to set the upper limit on admissible
+152 computational cost and to study the impact of sub-optimal proposal distribtions.
+153 Uniform proposal $( U )$ . The uniform distribution on the discrete interval from 1 to $K$ , i.e. $q ( z _ { i } | \bar { z } _ { i } ) : =$
+154 $q ( z _ { i } ) : = \mathrm { U n i f o r m } ( 1 , K )$ , is the simplest and computationally cheapest variant of the proposal
+155 distribution. However, due to poor mixing properties, the algorithm may converge slowly for high $K$ .
+156 Tabular proposal with forgetting $( T F )$ . The key requirement to design a proposal distribution is to
+157 restrict its computational complexity somewhere between that of the $\mathrm { U }$ and $\mathrm { o }$ proposals. One way to
+158 satisfy this constraint is to use the Markov chain, $( z _ { i , t } ) _ { t = 1 } ^ { T }$ , to learn a transition kernel, $p ( z _ { i } | \bar { z } _ { i } )$ , see,
+159 e.g. [3]. Unfortunately, this would require us to store a table with $K ^ { 2 }$ entries for each $i \in ( 1 , \ldots , N )$ ,
+160 161 which is very demanding evethe Markov chain and define: $q ( z _ { i } | \bar { z } _ { i } ) : = q _ { \alpha _ { i } } ( z _ { i } ) : = \mathcal { C } ( \alpha _ { i } )$ $K$ $N$ herefore, where $\mathcal { C } ( \pmb { \alpha } _ { i } ) \propto \Pi _ { k = 1 } ^ { K } \alpha _ { k , i } ^ { \bar { 1 ( } z _ { i } = k ) }$ nce inis the
+162 categorical distribution with the weights $\pmb { \alpha } _ { i } : = ( \alpha _ { 1 , i } , \dots , \alpha _ { K , i } )$ . For $\mathcal { L } ( \pmb { \alpha } _ { i } ) : = \Sigma _ { \tau = 1 } ^ { t } \log q _ { \pmb { \alpha } _ { i } } ( z _ { i , \tau } )$
+163 we obtain an estimate of $\alpha _ { i }$ at iteration $t$ as follows: $\begin{array} { r } { \alpha _ { i , t } : = \mathrm { \ a r g m a x } _ { \alpha _ { i } } \mathcal L ( \alpha _ { i } ) \ = \ \frac { n _ { i , t } } { t } } \end{array}$ t , with
+164 $n _ { i , t } = \Sigma _ { \tau = 1 } ^ { t } \mathbf { e } _ { z _ { i , t } }$ , where $\mathbf { e } _ { k }$ is the standard basis vector (a one-hot vector) with one at $k$ th position
+165 and zeros otherwise. This can be further rewritten into a recursive form: $n _ { i , t } = n _ { i , t - 1 } + \mathbf { e } _ { z _ { i , t } }$ or,
+166 using the Robbins-Monro step-size, $n _ { i , t } = ( { \bf 1 } - { \bf e } _ { z _ { i , t } } \gamma _ { t } ) \odot n _ { i , t - 1 } + \gamma _ { t } { \bf e } _ { z _ { i , t } }$ , where 1 is the vector of
+167 ones, and $\odot$ is the Hadamard product. We refer to this case simply as “table with forgetting” (TF)
+168 due to that it represents $N \times K$ table in the memory and $\gamma _ { t }$ is a forgetting factor.
+# 169 5 Related work
+Stochastic approximation expectation-maximization. The application of SA to prevent the evaluation
+1 of all $K$ components in mixture models has been overlooked for a long time. The reason is that the
+72 original motivation to combine the EM algorithm with SA is to address the analytical intractability
+73 of the expected value under $p _ { \theta } ( z | x )$ in (4), which is, however, almost always tractable for mixture
+74 models. The intractability issue is addressed by either the Monte Carlo SAEM (MCSAEM) [12]
+75 or the Markov chain Monte Carlo SAEM (MCMCSAEM) [31]. Applying the former approach to
+76 mixture models would be inefficient, since it evaluates $K$ joint distributions, $p _ { \theta } ( z , x )$ , before drawing
+177 $M$ samples from $p _ { \theta } ( z | x )$ . Therefore, this method reduces only the computational cost of updating the
+178 sufficient statistics. This is addressed by the latter approach, where $M < K$ samples from a proposal
+179 distribution, $q ( z | x )$ , is used to calculate $p _ { \theta } ( z , x )$ and also the sufficient statistics. However, all these
+180 methods process all data at every iteration, providing only a limited advantage over the conventional
+181 EM algorithm. Minibatch versions of these techniques have recently been proposed [27, 32, 1].
+182 All the above methods commonly assume $p _ { \theta } ( z , x )$ belonging to the exponential family. This provides
+183 a convenient, but limiting, property which allows (6) to be computed under a closed-form solution.
+184 The main contribution of our work is to release this restrictive assumption by admitting that $p _ { \theta } ( z , x )$
+185 (and thus $\mathcal { Q }$ ) is given by possibly complex and intractable transformations.
+186 Sparse and truncated variational techniques. There is only a small body of methods explicitly
+187 reducing the number of evaluated components. Their common aspect is that they follow from the
+188 variational framework, where the exact posterior, $p _ { \theta } ( z | x )$ , is approximated by a variational posterior,
+189 $q ( z | x )$ . This sparse, approximate, posterior is defined over a lower number of components, $M \ll K$ ,
+190 such that only the important components are selected, relying on relaxation of the hard EM algorithm
+191 from taking a single $M = 1$ assignment [26] to taking multiple $M \ll K$ assignments. The sparse
+192 SAEM (SSAEM) algorithm [24] selects the components by a quick partial sorting of the posterior
+193 probabilities, $p _ { \theta } ( z | x )$ . Again, this requires $K$ evaluations of $p _ { \theta } ( z , x )$ before the sorting, thus only
+194 reducing the amount of updated statistics. Similarly, the truncated SAEM (TSAEM) algorithm [18]
+195 selects $M < K$ cluster-to-cluster and $\bar { M } < K$ cluster-to-datapoint minimal Euclidean distances,
+196 preventing the problem in the SSAEM algorithm. However, all these distances are evaluated for all
+197 components in a pairwise manner, leading to $K ^ { 2 }$ -computational complexity, which makes the saving
+198 dubious. Similarly as before, these methods assume $p _ { \theta } ( z , x )$ to belong to the exponential family.
+![](images/0c516376ee7ecf27c7edaec6674fb19224d829bc10798ac37d421adee064aa0e.jpg)
+Figure 1: The training log-likelihood, $\mathcal { L } ( \boldsymbol { \theta } _ { t } )$ , versus the computational time (in seconds). Here, on the $\mathbf { X }$ -axis, the computational time at a current iteration, $t$ , is obtained by accumulating the time from the previous iterations. corresponds to $\mathcal { L } ( \boldsymbol { \theta } _ { t _ { 9 5 } } )$ , where $t _ { 9 5 }$ is the iteration of reaching $9 5 \%$ of max $\mathcal { L } ( \boldsymbol { \theta } _ { t } )$ . The projection of $^ { \circ }$ on the $\mathbf { X }$ -axis gives the time to reach $\mathcal { L } ( \boldsymbol { \theta } _ { t _ { 9 5 } } )$ . This experiment was performed with the following settings: $( D , \bar { K } , N , \omega , B , M , T ) = ( 1 0 , 1 0 0 , 1 0 k , 0 . 1 , \bar { 2 } 0 0 , 2 , 2 0 k )$ , see Section 6.1 for details. The results are averaged over five repetitions.
+199 We summarize the distinguishing features of the above discussed methods in Table 1.
+# 6 Experiments
+To demonstrate the key features of our algorithm—its low computational complexity, competitive learning performance, and generality—we use it below to train: (i) GMMs on synthetic datasets, and (ii) SPTNs [47] and (iii) mixtures of real NVP flows [48] on real datasets. All experiments have been performed on a Slurm cluster equipped with Intel Xeon Scalable Gold 6146 with 384GB of RAM.
+# 6.1 Gaussian mixture models
+Consider the special case of a data-generating distribution given by (1), with the components taking the form of the multivariate Gaussian distribution, $p _ { \eta _ { z } } ( x | z ) = \mathcal { N } ( x ; \mu _ { z } , \Sigma _ { z } )$ , where $\mu _ { z }$ is the mean value and $\Sigma _ { z }$ is the covariance matrix. The difficulty of learning GMMs heavily depends on the degree of interaction among all mixture components, hence having the ability to generate synthetic datasets with arbitrary overlap characteristics between all pairs of components is crucial for systematic
+![](images/0ced572977ae404a4e785379dc4987ce758b6f6930ca4d73adbe35955ed678b7.jpg)
+Figure 2: The absolute error, $\mathrm { A E } = | \mathcal { L } ( \theta _ { t _ { 9 5 } } ) - \mathcal { L } ( \theta ) |$ , versus the computational time (in seconds). All experiments use the following settings: $( D , K , \dot { N } , \omega , B , M , T ) = \bar { ( } 1 0 , 1 0 0 , 1 0 k , 0 . 1 , 2 0 0 , 2 , 2 0 k )$ , where the number of components, $K$ , (left), the batchsize, $B$ , (middle) and the number of samples, $M$ , (right) change for different values denoted by $( + , \sqsupset , \circ , \pmb { \triangle } )$ . At each of these points (marks), we perform an experiment as illustrated in Figure 1, find $\mathcal { L } ( \boldsymbol { \theta } _ { t _ { 9 5 } } )$ to compute the AE, and record the time corresponding to $t _ { 9 5 }$ . The results are averaged over five repetitions.
+211 evaluation of performance of learning algorithms [43]. Traditional techniques usually define overlap
+212 (or separation) of components only in terms of their mean vectors and maximum eigenvalues of the
+213 covariance matrices, not accounting for their rotation and mixing weights (see [36] for a detailed
+214 treatment of the problem). We therefore use a more objective measure of the clustering complexity
+215 defined by the total probability of misclassification [41], which allows to generate data with a
+216 user-defined degree of maximum pairwise overlap, $\omega$ .
+217 Experiment settings: We generate the parameters of (1), and the corresponding dataset, uniquely for a
+218 given quadruple $( D , K , N , \omega )$ . Therefore, the parameters of the generative model are known and we
+219 can measure and display the convergence of the training log-likelihood, $\mathcal { L } ( \boldsymbol { \theta } _ { t } )$ , compared to the exact
+220 log-likelihood, $\mathcal { L } ( \boldsymbol { \theta } ) \dot { }$ , for $t = ( 1 , \ldots , T )$ . We are further interested in the absolute error between the
+221 training log-likelihood at the iteration of reaching $9 5 \%$ of its maximum value, $t _ { 9 5 }$ , and the exact
+222 log-likelihood, i.e. $\mathrm { A E } = | \mathcal { L } ( \theta _ { t _ { 9 5 } } ) - \mathcal { L } ( \theta ) |$ .
+23 We also measure the computational time until reaching $t _ { 9 5 }$ . We have used $9 5 \%$ of the maximum
+24 value instead of the maximum value to prevent cases, where the model oscillate around target value,
+25 making the estimate of convergence time very noisy (for example MCSAEM in Figure 1).
+226 Algorithms: The GMMs belong to the exponential family of probability distributions. This allows us
+227 to find a closed-form, recursive, solution of (6), relying on a Robbins-Monro type of the step-size
+228 sequence, $( \gamma _ { t } ) _ { t = 1 } ^ { T }$ , [7, 44]. In this setting, we compare our MHSAEM algorithm with a number of
+229 related methods in Table 1. Note we use the acronyms U and TF to specify the proposal distribution of
+230 the MHSAEM algorithm (Section 4.3). However, we do not use the O-proposal, since the MHSAEM
+231 O algorithm is equivalent to the MCSAEM algorithm. All the SA-variants in Table 1 use a minibatch
+232 of size $B$ . The key quantity to reduce the number of evaluated components and/or sufficient statistics
+233 in the SSAEM, TSAEM, MCSAEM and MHSAEM algorithms is collectively denoted by $M$ (Section
+234 5). Note that we always keep $M = \bar { M }$ in the TSAEM algorithm (see Figure 1 and 2 for concrete
+235 numbers). We use the step-size given by $\gamma _ { t } = 1$ for $t = 1 , \ldots , 5 0$ and $\gamma _ { t } = 0 . 0 5$ otherwise. In
+236 this section, to counteract the issue of attaining poor local optima, we equip all algorithms with the
+237 anti-annealing schedule $( \beta _ { t } ) _ { t = 1 } ^ { T }$ , starting with $\beta _ { 1 } = 0 . 1$ , reaching $\beta _ { 2 / 3 T } = 1 . 2$ , and decreasing back
+238 to $\beta _ { T } = 1 . 0$ , see [43] for details. The initial estimates of: (i) $\mu _ { k }$ are uniformly drawn from the unit
+239 hyper-cube, (ii) $\Sigma _ { k }$ are fixed to unit diagonal matrix, and (iii) $\pi _ { k }$ are uniformly drawn from the unit
+240 interval (followed by normalization).
+241 Results: Figure 1 shows that the EM [13] and SAEM [44] algorithms take the longest time to
+242 converge, attaining a poor local optima. On the other hand, the MCSAEM [1] and MHSAEM (U
+243 and TF) algorithms achieve $\mathcal { L } ( \boldsymbol { \theta } _ { t _ { 9 5 } } )$ closest to the likelihood $\mathcal { L } ( \boldsymbol { \theta } )$ of the true model. Moreover, both
+244 MHSAEM algorithms reach this value in the shortest time compared to all the other methods. The
+245 SSAEM [24] and TSAEM [18] algorithms are comparable in terms of the computational time, but
+246 they both provide the lowest $\mathcal { L } ( \boldsymbol { \theta } _ { t _ { 9 5 } } )$ . In Figure 2, we investigate sensitivity of fitting the model to
+47 increasing values of $K$ , $B$ and $M$ by measuring the time and the likelihood again. In all the cases,
+48 the proposed MHSAEM algorithms achieve the lowest AE in the shortest time.
+SSAEM and TSAEM algorithms failed to converge for $M > 2$ and for $K > 5 0$ respectively. We believe this is caused by selecting only $M$ maximal probabilities in the SSAEM (or distances in the TSAEM) algorithm (Section 5), which prevents certain, but not a negligible number of, components from being updated, thus providing only a crude approximation of $\bar { p } _ { \theta } \bar { ( } z | x )$ . The results then suffer from substantial variational gap to the exact log-likelihood (Figure 1). On the contrary, MH sampler provides samples which consistently approximate $p _ { \theta } ( z | x )$ despite evaluating much lower number of components in each step.
+# 6.2 Sum-product transform networks
+The sum product networks (SPNs) are a deep learning extension of finite mixture models. They can be interpreted as a mixture of trees [60], where each tree corresponds to a component. Therefore, they can be cast into the form of (1), but the number of components grows exponentially with their depth. In this section, we use recently proposed SPTNs which introduce additional transformation nodes to provide better expressiveness than the SPNs (SPTNs effectively generalize SPNs and flow models into one large family of models).
+Experimental settings: We use 19 real datasets from the UCI database [16, 37, 35, 54], preprocessed in the same way as in [46]. For each experiment, we randomly split the data into $64 \%$ , $16 \%$ and $20 \%$ for training, validation and testing, respectively. We calculate the average log-likelihood on the test set and measure again the time to reach $9 5 \%$ of the maximal training log-likelihood, $\mathcal { L } ( \boldsymbol { \theta } _ { t _ { 9 5 } } )$ .
+To evaluate various (possibly shallow and/or deep) architectures of SPTNs, we fit each dataset with all the following combinations of hyper-parameters1: $s \in ( 8 , 3 2 , 1 2 8 )$ , $b \in ( 2 , 4 , 6 , 8 )$ , $l \in ( 2 , 3 , 4 )$ , where $s$ is the number of children of each sum node, $b$ is the number of partitions of each product node, and $l$ is the number of layers (one layer contains sum and product nodes). The number of components of the SPTN, after its conversion into (1), is given as follows: $K = s ^ { l }$ . Note that the maximum number of components for the investigated parameters of the SPTN is 268,435,456. To reduce the space of possible architectures, we restrict ourselves only to (i) the leaf nodes given by $\mathcal { N } ( 0 , \bf { I } )$ ; (ii) affine transformations fixed to the singular value decomposition, choosing the the Givens parameterization for the unitary matrices [47]; and (iii) no sharing of any type of nodes [47].
+276 Algorithms: We evaluate only on the MHSAEM-U algorithm—due to its favourable computational
+277 complexity and simplicity—and compare it with the stochastic gradient-descent (SGD) algorithm,
+278 which is routinely used to train SP(T)Ns [45, 47]. In this case, SGD in each iteration performs
+279 computations over all subtrees of the network, whereas the MHSAEM-U algorithm computes with
+280 only $M = 1$ subtrees, thus we should observe speed-up of the computations. In our implementation,
+281 both these methods perform optimization of their respective objective functions—the log-likelihood
+282 (2) for SGD and the EM objective (8) for MHSAEM-U—via the use of the automatic differentiation
+283 and the ADAM optimizer [29], using $B = 1 0 0$ and $T = 2 0 0 0 0$ .
+Results: Since each dataset might benefit from a different architecture, Table 6.2 shows the test log-likelihood of the architectures selected according to the best likelihood measured on the validation set and the corresponding speed-up. The test log-likelihoods reveal that the MHSAEM-U algorithm outperforms the SGD algorithm on 10 out of 19 datasets, which was not originally the goal, but the added stochasticity helps to escape poor local minima. The speed-up demonstrates lower computational complexity of the MHSAEM-U algorithm on 17 out of 19 datasets, which was the main goal. The magic-telescope and wine datasets show approximately $1 0 2 \times$ and $7 5 \times$ speed-up, respectively, while on very small datasets (pima-indians and iris), the SGD is faster due to effective implementation. In the supplementary material, we present Table 3, exhibiting the same trends on a fixed architecture.
+# 6.3 Mixtures of real NVP flows
+We consider another class of mixture models (1), where each component $p _ { \eta _ { z } } ( x | z )$ is transformed by the flow model—real NVP [15]. These transformations are parameterized via deep neural networks, allowing for flexible adjustment of the learning capacity of each component.
+Table 2: The speed-up and test log-likelihood, $\mathcal { L } ^ { \mathrm { t e s t } }$ , for the SGD and MHSAEM-U algorithms. The test log-likelihood (higher is better) is computed for the best model, with the corresponding $K$ , which is selected based on the validation log-likelihood. The speed-up is computed as the ratio of MHSAEM-U to SGD, i.e. their time to reach $9 5 \%$ of the training log-likelihood. The results are averaged over five repetitions. Then, the higher test log-likelihood is highlighted with bold blue, and and no speed-up is highlighted with red. The average rank is computed as the standard competition (“1224”) ranking [14] on each dataset (lower is better).
+<table><tr><td rowspan="3"></td><td colspan="5">Sum-product transformnetworks</td><td colspan="5">Mixtures of real NVP flows SGD</td></tr><tr><td rowspan="2"></td><td colspan="2">SGD</td><td colspan="2">MHSAEM-U</td><td colspan="2"></td><td colspan="2"></td><td colspan="2">MHSAEM-U</td></tr><tr><td>speed-up</td><td>Ltest</td><td>K</td><td>Ltest</td><td>K</td><td>speed-up</td><td>Ltest</td><td>K 32</td><td>Ltest</td><td>K</td></tr><tr><td>breast-cancer-wisconsin</td><td>4.66</td><td>-4.66</td><td>64</td><td>1.43</td><td>1024</td><td>0.63</td><td>-99.85</td><td></td><td>-39.31</td><td></td><td>128</td></tr><tr><td>cardiotocography</td><td>10.55</td><td>59.52</td><td>512</td><td>31.04</td><td>1024</td><td></td><td>9.85</td><td>54.34</td><td>32</td><td>56.08</td><td>128</td></tr><tr><td>magic-telescope</td><td>102.53</td><td>-3.65</td><td>512</td><td>-5.03</td><td>1024</td><td></td><td>3.74</td><td>-3.97</td><td>8</td><td>-4.22</td><td>8</td></tr><tr><td>pendigits</td><td>4.89</td><td>0.88</td><td>1024</td><td>-4.86</td><td>16384</td><td></td><td>4.17</td><td>1.46</td><td>8</td><td>0.48</td><td>8</td></tr><tr><td>pima-indians</td><td>0.37</td><td>-8.54</td><td>64</td><td>-7.62</td><td></td><td>64</td><td>1.35</td><td>-20.09</td><td>128</td><td>-16.33</td><td>128</td></tr><tr><td>wall-following-robot</td><td>3.43</td><td>1.84</td><td>1024</td><td>-11.3</td><td>16384</td><td></td><td>22.21</td><td>-14.26</td><td>128</td><td>-17.56</td><td>128</td></tr><tr><td>waveform-1</td><td>4.35</td><td>-26.14</td><td>64</td><td>-23.91</td><td>1024</td><td></td><td>3.72</td><td>-34.12</td><td>8</td><td>-33.42</td><td>8</td></tr><tr><td>waveform-2</td><td>4.82</td><td>-26.21</td><td>64</td><td>-23.91</td><td></td><td>1024</td><td>4.12</td><td>-34.15</td><td>8</td><td>-33.64</td><td>8</td></tr><tr><td>yeast</td><td>20.57</td><td>10.26</td><td>512</td><td>5.18</td><td>1024</td><td></td><td>14.49</td><td>6.61</td><td>128</td><td>9.59</td><td>128</td></tr><tr><td>ecoli</td><td>1.86</td><td>-5.5</td><td>64</td><td>-0.22</td><td>1024</td><td></td><td>2.15</td><td>-11.37</td><td>128</td><td>-10.64</td><td>128</td></tr><tr><td>ionosphere</td><td>1.88</td><td>-20.27</td><td>64</td><td>-5.93</td><td></td><td>512</td><td>2.74</td><td>-87.01</td><td>128</td><td>-42.75</td><td>128</td></tr><tr><td>iris</td><td>0.23</td><td>-10.65</td><td>64</td><td>-1.49</td><td>16384</td><td></td><td>3.28</td><td>-16.34</td><td>128</td><td>-9.21</td><td>32</td></tr><tr><td>page-blocks</td><td>12.18</td><td>12.21</td><td>512</td><td>6.84</td><td>1024</td><td></td><td>44.95</td><td>17.13</td><td>128</td><td>17.94</td><td>32</td></tr><tr><td>parkinsons</td><td>1.46</td><td>-21.85</td><td>64</td><td>0.5</td><td></td><td>512</td><td>3.09</td><td>-566.58</td><td>128</td><td>-33.31</td><td>32</td></tr><tr><td>sonar</td><td>2.96</td><td>-95.39</td><td>512</td><td>-69.29</td><td></td><td>64</td><td>2.52</td><td>-622.2</td><td>128</td><td>-88.81</td><td>128</td></tr><tr><td>statlog-segment</td><td>1.44</td><td>47.35</td><td>512</td><td>26.53</td><td>16384</td><td></td><td>38.49</td><td>35.84</td><td>128</td><td>42.04</td><td>32</td></tr><tr><td>statlog-vehicle</td><td>2.97</td><td>-4.25</td><td>64</td><td>-5.45</td><td>1024</td><td></td><td>6.78</td><td>-31.34</td><td>32</td><td>-26.43</td><td>128</td></tr><tr><td>wine rank</td><td>75.42</td><td>-25.99</td><td>1024</td><td>-13.27</td><td></td><td>1024</td><td>2.05</td><td>-171.58</td><td>128</td><td>-25.57</td><td>128</td></tr><tr><td></td><td></td><td>1.56</td><td></td><td>1.44</td><td></td><td></td><td></td><td>1.83</td><td></td><td>1.17</td><td></td></tr></table>
+298 Experimental settings: We use the same experimental settings and evaluation metrics as in Section
+299 6.2. We apply the mixture model on all datasets, changing the number of components as follows:
+300 $K \in ( 8 , 3 2 , 1 2 8 )$ . Each real NVP-based component in the mixture model has (i) the translation
+301 function parameterized via multi-layer perceptron with a single hidden layer of dimension 10, using
+302 the rectified linear activation function; and (ii) the scale function parameterized via the same network
+303 except with the hyperbolic tangent activation function. We do not use the batch normalization [15] and
+304 we stack two layers of the translation-scale transformation (we have used implementation from [20]).
+Algorithms: The algorithms and their settings are the same as those in Section 6.2.
+306 Results: The experimental results are presented in right part of Table 6.2. They are similar to those
+307 obtained in the previous section. In terms of the test log-likelihood, the MHSAEM-U algorithm
+308 outperforms the SGD algorithm on all but three datasets, and it provides a substantial speed-up on all
+309 datasets except one. The test likelihood of models with the real NVP flows is most of the time worse
+310 than that of SPTNs with the affine transformations. As explained in the supplementary, this is due to
+311 the overfitting, which has been observed in [47].
+# 312 7 Conclusion
+313 This paper has presented a method to decrease computational complexity of fitting mixture models,
+314 including their generalizations, such as sum-product-(transform) networks and mixtures of flow
+315 models. The speed-up is achieved by evaluating and updating only a single component (per iteration),
+316 where the Metropolis-Hasting algorithm ensures sampling of components from a proper posterior. An
+317 experimental comparison on all three classes of models mentioned above confirmed the theoretical
+318 expectations. The method significantly speeds-up the fitting time and, importantly, without sacrificing
+319 the quality of the fit. In fact, the likelihood was better than that of the models fitted by the EM
+320 algorithm or the SGD algorithm in more than $50 \%$ of cases. We attribute this to higher stochasticity,
+321 which helps to escape from poor local minima.
+322 In the experiments, the proposed method has used a uniform proposal distribution in the MH sampler.
+323 Despite outperforming the alternative methods, we conjecture that this limits the speed of convergence.
+324 Therefore, we believe that there is still a room for improvement in the implementation. We plan to
+325 address these issues in future work.
+The presented method decreases the computational complexity of fitting large (and deep) mixture models, which leads to five to hundred time speed-up depending on a size of the problem (although negative exceptions occurs). We believe this line of research, which we want to continue, to have important benefits. First, it is directly related to decrease in energy consumption and in production of CO2 (we expect similar rates as the speedup). Second, it has a positive effect on financial aspects of deploying (and experimenting with) mixture models. Third, it decreases the hardware requirements, as in all experiments presented above the model was fitted on a single-core.
+# References
+[1] S. Allassonnière and J. Chevallier. A new class of stochastic EM algorithms: Escaping local maxima and handling intractable sampling. Computational Statistics & Data Analysis, 159:107159, 2021.
+[2] S. Ayesha, M. K. Hanif, and R. Talib. Overview and comparative study of dimensionality reduction techniques for high dimensional data. Information Fusion, 59:44–58, 2020.
+[3] D. S. Bai. Efficient estimation of transition probabilities in a Markov chain. The Annals of Statistics, pages 1305–1317, 1975.
+[4] D. M. Blei, A. Kucukelbir, and J. D. McAuliffe. Variational inference: A review for statisticians. Journal of the American statistical Association, 112(518):859–877, 2017.
+[5] J. Blömer and K. Bujna. Adaptive seeding for Gaussian mixture models. In Pacific-asia conference on knowledge discovery and data mining, pages 296–308. Springer, 2016.
+[6] T. Campbell and B. Beronov. Sparse variational inference: Bayesian coresets from scratch. arXiv preprint arXiv:1906.03329, 2019.
+[7] O. Cappé and E. Moulines. On-line expectation–maximization algorithm for latent data models. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 71(3):593–613, 2009.
+[8] J. Chen, J. Zhu, Y. Teh, and T. Zhang. Stochastic expectation maximization with variance reduction. Advances in Neural Information Processing Systems, page 7967, 2018.
+[9] N. Chopin. A sequential particle filter method for static models. Biometrika, 89(3):539–552, 2002.
+[10] G. Compiani and Y. Kitamura. Using mixtures in econometric models: A brief review and some new results. The Econometrics Journal, 19(3):C95–C127, 2016.
+[11] S. Dasgupta. Learning mixtures of Gaussians. In 40th Annual Symposium on Foundations of Computer Science (Cat. No. 99CB37039), pages 634–644. IEEE, 1999.
+[12] B. Delyon, M. Lavielle, E. Moulines, et al. Convergence of a stochastic approximation version of the EM algorithm. The Annals of Statistics, 27(1):94–128, 1999.
+[13] A. P. Dempster, N. M. Laird, and D. B. Rubin. Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society: Series B (Methodological), 39(1):1–22, 1977.
+[14] J. Demšar. Statistical comparisons of classifiers over multiple data sets. The Journal of Machine Learning Research, 7:1–30, 2006.
+[15] L. Dinh, J. Sohl-Dickstein, and S. Bengio. Density estimation using real NVP. In 5th International Conference on Learning Representations, ICLR 2017, 2017.
+[16] D. Dua and C. Graff. UCI machine learning repository, 2017.
+[17] D. Feldman, M. Faulkner, and A. Krause. Scalable training of mixture models via coresets. In Proceedings of the 24th International Conference on Neural Information Processing Systems, pages 2142–2150, 2011. 372 [18] D. Forster and J. Lücke. Can clustering scale sublinearly with its clusters? A variational EM acceleration of GMMs and k-means. In International Conference on Artificial Intelligence and Statistics, pages 124–132. PMLR, 2018. [19] G. Fort, E. Moulines, et al. Convergence of the Monte Carlo expectation maximization for curved exponential families. Annals of Statistics, 31(4):1220–1259, 2003. [20] J. Franc˚u. Continuousflows.jl. https://github.com/janfrancu/ContinuousFlows.jl, 2020. [21] K. Greff, S. van Steenkiste, and J. Schmidhuber. Neural expectation maximization. In Proceedings of the 31st International Conference on Neural Information Processing Systems, pages 6694–6704, 2017. [22] R. Gribonval, A. Chatalic, N. Keriven, V. Schellekens, L. Jacques, and P. Schniter. Sketching datasets for large-scale learning (long version). arXiv preprint arXiv:2008.01839, 2020. [23] M. D. Hoffman, D. M. Blei, C. Wang, and J. Paisley. Stochastic variational inference. Journal of Machine Learning Research, 14(5), 2013. [24] M. C. Hughes and E. B. Sudderth. Fast learning of clusters and topics via sparse posteriors. arXiv preprint arXiv:1609.07521, 2016. [25] K. Humphreys and D. Titterington. Approximate Bayesian inference for simple mixtures. In COMPSTAT, pages 331–336. Springer, 2000. [26] B.-H. Juang and L. R. Rabiner. The segmental K-means algorithm for estimating parameters of hidden Markov models. IEEE Transactions on acoustics, speech, and signal Processing, 38(9):1639–1641, 1990. [27] B. Karimi, M. Lavielle, and É. Moulines. On the convergence properties of the mini-batch EM and MCEM algorithms, 2019. [28] N. Keriven, A. Bourrier, R. Gribonval, and P. Pérez. Sketching for large-scale learning of mixture models. Information and Inference: A Journal of the IMA, 7(3):447–508, 2018. [29] D. P. Kingma and J. Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014. [30] A. Kucukelbir, D. Tran, R. Ranganath, A. Gelman, and D. M. Blei. Automatic differentiation variational inference. The Journal of Machine Learning Research, 18(1):430–474, 2017. [31] E. Kuhn and M. Lavielle. Coupling a stochastic approximation version of EM with an MCMC procedure. ESAIM: Probability and Statistics, 8:115–131, 2004. [32] E. Kuhn, C. Matias, and T. Rebafka. Properties of the stochastic approximation EM algorithm with mini-batch sampling. Statistics and Computing, 30(6):1725–1739, 2020. [33] A. Lagrange, M. Fauvel, and M. Grizonnet. Large-scale feature selection with Gaussian mixture models for the classification of high dimensional remote sensing images. IEEE Transactions on Computational Imaging, 3(2):230–242, 2017. [34] M. Lavine and M. West. A Bayesian method for classification and discrimination. Canadian Journal of Statistics, 20(4):451–461, 1992. [35] M. Little, P. McSharry, S. Roberts, D. Costello, and I. Moroz. Exploiting nonlinear recurrence and fractal scaling properties for voice disorder detection. Nature Precedings, pages 1–1, 2007. [36] R. Maitra and V. Melnykov. Simulating data to study performance of finite mixture modeling and clustering algorithms. Journal of Computational and Graphical Statistics, 19(2):354–376, 2010.
+15 [37] O. L. Mangasarian and W. H. Wolberg. Cancer diagnosis via linear programming. Technical report, University of Wisconsin-Madison Department of Computer Sciences, 1990.
+17 [38] J.-M. Marin, K. Mengersen, and C. P. Robert. Bayesian modelling and inference on mixtures of distributions. Handbook of statistics, 25:459–507, 2005.
+19 [39] C. A. McGrory, D. C. Ahfock, J. A. Horsley, and C. L. Alston. Weighted Gibbs sampling for mixture modelling of massive datasets via coresets. Stat, 3(1):291–299, 2014. [40] G. J. McLachlan, S. X. Lee, and S. I. Rathnayake. Finite mixture models. Annual review of statistics and its application, 6:355–378, 2019.
+423 [41] V. Melnykov, W.-C. Chen, and R. Maitra. MixSim: An R package for simulating data to study performance of clustering algorithms. Journal of Statistical Software, 51(12):1, 2012.
+25 [42] T. Monnier, T. Groueix, and M. Aubry. Deep transformation-invariant clustering. In Conference on Neural Information Processing Systems (NeurIPS 2020), 2020. [43] I. Naim and D. Gildea. Convergence of the EM algorithm for Gaussian mixtures with unbalanced mixing coefficients. In Proceedings of the 29th International Coference on International Conference on Machine Learning, pages 1427–1431, 2012. [44] H. D. Nguyen, F. Forbes, and G. J. McLachlan. Mini-batch learning of exponential family finite mixture models. Statistics and Computing, pages 1–18, 2020. [45] R. Peharz, A. Vergari, K. Stelzner, A. Molina, X. Shao, M. Trapp, K. Kersting, and Z. Ghahramani. Random sum-product networks: A simple and effective approach to probabilistic deep learning. In Uncertainty in Artificial Intelligence, pages 334–344. PMLR, 2020. [46] T. Pevný. Loda: Lightweight on-line detector of anomalies. Machine Learning, 102(2):275–304, 2016. [47] T. Pevný, V. Šmídl, M. Trapp, O. Polácek, and T. Oberhuber. Sum-product-transform networks: ˇ Exploiting symmetries using invertible transformations. arXiv preprint arXiv:2005.01297, 2020. [48] G. G. Pires and M. A. Figueiredo. Variational mixture of normalizing flows. arXiv preprint arXiv:2009.00585, 2020.
+42 [49] A. Rau, C. Maugis-Rabusseau, M.-L. Martin-Magniette, and G. Celeux. Co-expression analysis of high-throughput transcriptome sequencing data with Poisson mixture models. Bioinformatics, 31(9):1420–1427, 2015.
+45 [50] R. A. Redner and H. F. Walker. Mixture densities, maximum likelihood and the EM algorithm. SIAM review, 26(2):195–239, 1984.
+47 [51] H. Robbins and S. Monro. A stochastic approximation method. The annals of mathematical statistics, pages 400–407, 1951.
+49 [52] C. Robert and G. Casella. Monte Carlo statistical methods. Springer Science & Business Media, 2013. [53] W. Siblini, P. Kuntz, and F. Meyer. A review on dimensionality reduction for multi-label classification. IEEE Transactions on Knowledge and Data Engineering, 2019.
+53 [54] J. P. Siebert. Vehicle recognition using rule based methods. 1987. [55] C. Viroli and G. J. McLachlan. Deep gaussian mixture models. Statistics and Computing, 29(1):43–51, 2019.
+56 [56] C. F. J. Wu. On the convergence properties of the EM algorithm. The Annals of statistics, pages 95–103, 1983.
+58 [57] D. Wu and J. Ma. An effective EM algorithm for mixtures of Gaussian processes via the MCMC sampling and approximation. Neurocomputing, 331:366–374, 2019.
+60 [58] W. Xiang, A. Karfoul, C. Yang, H. Shu, and R. L. B. Jeannès. An exact line search scheme to
+61 accelerate the EM algorithm: Application to Gaussian mixture models identification. Journal of
+62 computational science, 41:101073, 2020.
+63 [59] M. Zhang, Y. Fu, K. M. Bennett, and T. Wu. Computational efficient variational Bayesian Gaus
+64 sian mixture models via coreset. In 2016 International Conference on Computer, Information
+65 and Telecommunication Systems (CITS), pages 1–5. IEEE, 2016.
+66 [60] H. Zhao, P. Poupart, and G. Gordon. A unified approach for learning the parameters of sum
+67 product networks. In Proceedings of the 30th International Conference on Neural Information
+68 Processing Systems, pages 433–441, 2016.
+1. For all authors...
+(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]
+(b) Did you describe the limitations of your work? [Yes] Our main contribution is computational speedup. Cases where it was not achieved are highlighted in the experimental section.
+(c) Did you discuss any potential negative societal impacts of your work? [No] We do not foresee any potential negative impact.
+(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes]
+2. If you are including theoretical results...
+(a) Did you state the full set of assumptions of all theoretical results? [N/A] (b) Did you include complete proofs of all theoretical results? [N/A]
+3. If you ran experiments...
+(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] The code is available in a github repository. All dataset are public from the UCI database.
+(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] See Section 6.
+(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [No] We report only average of Monte Carlo repetitions, the error bars were too small to have any visual impact in the reported logarithmic scale.
+(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes]
+4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets...
+(a) If your work uses existing assets, did you cite the creators? [Yes] We use 20 datasets from UCI, we cite the required papers for each dataset, mostly the UCI database and few additional publications.
+(b) Did you mention the license of the assets? [No] The data are publically available, we comply with the requirement on citing appropriate publications.
+(c) Did you include any new assets either in the supplemental material or as a URL? [No]
+(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A]
+(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A]
+5. If you used crowdsourcing or conducted research with human subjects...
+(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]
+(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]
+(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A]

parse/train/ZS394D3djsg/ZS394D3djsg_content_list.json ADDED Viewed

	@@ -0,0 +1,1195 @@

+[
+    {
+        "type": "text",
+        "text": "Fitting large mixture models using stochastic component selection ",
+        "text_level": 1,
+        "bbox": [
+            228,
+            122,
+            769,
+            172
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Anonymous Author(s)   \nAffiliation   \nAddress   \nemail ",
+        "bbox": [
+            423,
+            226,
+            580,
+            281
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "Abstract ",
+        "text_level": 1,
+        "bbox": [
+            462,
+            318,
+            535,
+            334
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "1 Traditional methods for unsupervised learning of finite mixture models require to   \n2 evaluate the likelihood of all components of the mixture. This becomes computa  \n3 tionally prohibitive when the number of components is large, as it is, for example,   \n4 in the sum-product (transform) networks. As a remedy, we propose an approach   \n5 combining the expectation maximization and the Metropolis-Hastings algorithm   \n6 to evaluate only a small number of, stochastically sampled, components, thus   \n7 substantially reducing the computational cost. We put emphasis on generality of   \n8 our method, equipping it with the ability to train both shallow and deep mixture   \n9 models which involve complex, and possibly nonlinear, transformations. The   \n10 performance of our method is illustrated in a variety of synthetic and real-data   \n11 contexts, considering deep models, such as mixtures of normalizing flows and   \n12 sum-product (transform) networks. ",
+        "bbox": [
+            148,
+            348,
+            766,
+            515
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "13 1 Introduction ",
+        "text_level": 1,
+        "bbox": [
+            148,
+            539,
+            312,
+            556
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "14 Finite mixture models [40] constitute a fundamental class of density estimation models. They   \n15 have been successfully applied in diverse fields, including bioinformatics [49], econometrics [10],   \n16 engineering [33], etc. A mixture model relies on a weighted sum of probability distributions—here   \n17 referred to as components—to cluster $N$ unlabelled datapoints into $K$ categories. The traditional   \n18 maximum likelihood techniques train the model by optimizing either (i) the marginal likelihood via   \n19 gradient-descent [50] or (ii) the evidence lower bound via variational methods [4], including the   \n20 expectation-maximization (EM) [13]. The dependence structure among approximate, variational,   \n21 distributions then ranges from the fully independent (mean-field) [25] to fully dependent [30]. The   \n22 sampling-based techniques target the posterior distribution using sequential Monte Carlo [9] or   \n23 Markov chain Monte Carlo [52], e.g. via the Gibbs [34] or Metropolis-Hastings sampling [38]. The   \n24 computational cost of these methods typically scales with $\\mathcal { O } ( T K N D )$ operations, where $N$ and $K$   \n25 are defined above, $T$ is the number of iterations and $D$ is the dimension of data.   \n26 Various methods to decrease the computational cost via any factor in $\\mathcal { O } ( T K N D )$ have been proposed.   \n27 $T$ can be lowered by proper initialization, e.g. the optimal seeding [5]; an efficient step-size schedule,   \n28 e.g. the line-search [58]; or increased estimation precision, e.g. the variance reduction [8]. $N$ is often   \n29 reduced using the coreset methods, which approximate the original dataset by a weighted dataset   \n30 such that the exact and approximate marginal likelihoods are close. The weighted variants of the   \n31 variational [17, 59, 6] and sampling-based [39] methods then process the coresets. Reducing $D$ relies   \n32 on the compression of data into smaller representations via random projections [53, 2], which is   \n33 achieved in two ways: (i) each data item is projected into an individual representation [11]; (ii) all   \n34 data items are projected into an overall representation, commonly referred to as sketch [28, 22].   \n35 Nevertheless, all the aforementioned techniques—including those with reduced computational cost—   \n36 evaluate all $K$ components. This is very demanding for large models, and the problem is even more   \n37 severe for mixtures involving intricate models, such as neural networks [21, 42], Gaussian processes   \n38 [57], normalizing flows [48]; or deep mixtures, including sum-product (transform) networks [45, 47],   \n39 deep Gaussian mixture models [55], etc. In spite of this, a little attention has been paid to the design   \n40 of algorithms which does not evaluate all $K$ components. The notable exceptions are the sparse EM   \n41 algorithm [24] and the truncated variational EM algorithm [18], see Table 1 and Section 5 for details.   \n42 Moreover, the methods are mostly tailored for a specific class of mixture models, e.g. the Gaussian   \n43 mixture models. ",
+        "bbox": [
+            147,
+            570,
+            825,
+            736
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            145,
+            742,
+            825,
+            867
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            148,
+            873,
+            828,
+            901
+        ],
+        "page_idx": 0
+    },
+    {
+        "type": "table",
+        "img_path": "images/5e346a33d4bf96097d66d042d2edb3a796cd7475f2111dc5b8813d4424db6d24.jpg",
+        "table_caption": [
+            "Table 1: The computational features of various EM algorithms. We compare whether the methods (i) perform the computations with a reduced number of data (minibatching), (ii) update a lower number of statistics, (iii) make less evaluations of the conditional likelihood, and (iv) are suitable for training of deep models. Here, EM, SA, S, T, MC and MH stand for expectation-maximization, stochastic approximation, sparse, truncated, Monte Carlo and Metropolis-Hastings, respectively. "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td>Feature/Algorithm</td><td>EM [13]</td><td>SAEM [44]</td><td>SSAEM [24]</td><td>TSAEM [18]</td><td>MCSAEM [1]</td><td>MHSAEM (ours)</td></tr><tr><td>B&lt;Ndatapoints</td><td>×</td><td></td><td></td><td>√</td><td></td><td></td></tr><tr><td>M&lt;K statistics</td><td>xx</td><td></td><td></td><td></td><td>厂</td><td></td></tr><tr><td>M&lt;Klikelihoods</td><td></td><td>xx</td><td>×</td><td></td><td>X</td><td></td></tr><tr><td>deep models</td><td>×</td><td>×</td><td>×</td><td>X</td><td>×</td><td></td></tr></table>",
+        "bbox": [
+            222,
+            165,
+            774,
+            244
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            147,
+            273,
+            826,
+            372
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "44 In this paper, we make the following contributions: ",
+        "bbox": [
+            147,
+            377,
+            509,
+            392
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "45 • We propose an EM-based algorithm which relies on the MH sampler to stochastically evaluate less   \n46 components in mixture models, substantially reducing the computational cost.   \n7 • We design our method to enable optimization of fairly generic EM objective functions, making it   \n48 suitable for training of both shallow and deep mixture models.   \n49 • We apply our approach to Gaussian mixture mdoels (GMMs) and their generalizations: sum  \n50 product-transform networks (SPTNs) and mixtures of real-valued non-volume preserving (real   \n51 NVP) flows [15], reaching approximately $1 0 0 \\times$ speed-up compared to state-of-the-art methods. ",
+        "bbox": [
+            151,
+            405,
+            825,
+            434
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            155,
+            441,
+            823,
+            472
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            148,
+            479,
+            825,
+            522
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "52 2 Problem formulation ",
+        "text_level": 1,
+        "bbox": [
+            148,
+            553,
+            382,
+            570
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "A finite mixture model characterizes the relation between an observed (known) variable, 53 $\\boldsymbol { x } \\in \\times \\subseteq \\mathbb { R } ^ { D }$ , 54 and a latent (unknown) variable, $z \\in Z : = \\{ 1 , \\dots , K \\}$ , via the marginal (incomplete-data) likelihood 55 in the following form: ",
+        "bbox": [
+            147,
+            587,
+            825,
+            630
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/1fd8c64bc931f74af8171cacb46094a19b2d53d8b9408dc1581f5a44dfea91a9.jpg",
+        "text": "$$\np _ { \\theta } ( x ) = \\sum _ { k = 1 } ^ { K } p _ { \\eta _ { k } } ( x | z = k ) p _ { \\pi _ { k } } ( z = k ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            369,
+            633,
+            627,
+            678
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "56 where $\\theta : = ( \\pi _ { 1 } , \\eta _ { 1 } , \\dots , \\pi _ { K } , \\eta _ { K } ) \\in \\Theta$ are unknown parameters. Here, $\\eta _ { z }$ are the parameters of the 57 conditional likelihood, $p _ { \\eta _ { z } } ( x | z )$ , and $\\pi _ { z }$ is the weight which parameterizes the prior, $p _ { \\pi _ { z } } ( z ) = \\pi _ { z }$ , and satisfies 58 $0 \\leq \\pi _ { k } \\leq 1$ for each $k \\in { \\mathord { \\mathbb { Z } } }$ and $\\textstyle \\sum _ { k = 1 } ^ { K } \\pi _ { k } = 1$ . ",
+        "bbox": [
+            145,
+            685,
+            826,
+            734
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "59 Given a set of independent and identically distributed data, $\\mathbf { x } : = ( x _ { i } ) _ { i = 1 } ^ { N }$ , our goal is to learn the   \n60 unknown parameters of the marginal log-likelihood, ",
+        "bbox": [
+            148,
+            739,
+            825,
+            770
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "equation",
+        "img_path": "images/3a533a8160c37e7b9fa1209ffe45b29c0eb14f6404e4bc9e51f2458ababc808c.jpg",
+        "text": "$$\n\\mathcal { L } ( \\theta ) : = \\log p _ { \\theta } ( \\mathbf { x } ) = \\sum _ { i = 1 } ^ { N } \\log \\sum _ { k = 1 } ^ { K } p _ { \\eta _ { k } } ( x _ { i } | z _ { i } = k ) p _ { \\pi _ { k } } ( z _ { i } = k ) .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            295,
+            780,
+            700,
+            824
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "61 The marginalization in (2) is tractable for almost all forms of $p _ { \\eta _ { z } } ( x | z )$ . Indeed, we consider $p _ { \\eta _ { z } } ( x | z )$   \n62 to belong to an arbitrary family of $\\eta _ { z }$ -differentiable probability distributions. However, we assume that   \n63 $K$ is high, making the marginalization in (2) computationally costly, thus rendering the optimization   \n64 objective presumably intractable. Therefore, we want to design a computationally efficient algorithm,   \n65 requiring only $M < K$ evaluations of $p _ { \\eta _ { z } } ( x | z )$ at each iteration.   \n67 The maximum likelihood estimation seeks the parameters maximizing the marginal log-likelihood,   \n68 $\\theta ^ { M L } : = \\arg \\operatorname* { m a x } _ { \\theta \\in \\Theta } \\mathcal { L } ( \\theta )$ . The traditional EM algorithm [13] addresses this task indirectly, i.e. by   \n69 optimizing the evidence lower bound (ELBO), ",
+        "bbox": [
+            147,
+            840,
+            826,
+            912
+        ],
+        "page_idx": 1
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            147,
+            121,
+            825,
+            162
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/df55efbf1dd9a5a5ef40f11a29cb92a7c818d04135e3c849687eb5724cd486b5.jpg",
+        "text": "$$\n\\mathcal { L } ( \\theta ) \\geq \\mathcal { Q } ( \\theta ) + \\mathcal { H } ( \\hat { \\theta } ) : = \\mathrm { E L B O } ( \\hat { \\theta } ) ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            379,
+            170,
+            619,
+            189
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "where 70 $\\mathcal { H } ( \\hat { \\theta } ) : = - \\mathsf E _ { p _ { \\hat { \\theta } } ( \\mathbf { z } | \\mathbf { x } ) } [ \\log p _ { \\hat { \\theta } } ( \\mathbf { z } | \\mathbf { x } ) ]$ is the differential entropy at an estimate, $\\hat { \\theta } \\in \\Theta$ , and ",
+        "bbox": [
+            140,
+            198,
+            771,
+            215
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/8dc49814f958acdf8ce0e11b963a6e6101e3e5d1439603310c411ba01588b3e9.jpg",
+        "text": "$$\n\\mathcal { Q } ( \\theta ) : = \\mathsf { E } _ { p _ { \\hat { \\theta } } ( \\mathbf { z } | \\mathbf { x } ) } [ \\log p _ { \\theta } ( \\mathbf { z } , \\mathbf { x } ) ] = \\sum _ { i = 1 } ^ { N } \\sum _ { k = 1 } ^ { K } p _ { \\theta } ( z _ { i } = k | x _ { i } ) \\log p _ { \\theta } ( z _ { i } = k , x _ { i } )\n$$",
+        "text_format": "latex",
+        "bbox": [
+            254,
+            222,
+            743,
+            267
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "71 is the EM objective function. Here, $p _ { \\theta } ( \\mathbf { z } , \\mathbf { x } )$ is the joint (complete-data) likelihood, and $p _ { \\theta } ( \\mathbf { z } | \\mathbf { x } )$ is   \n72 the posterior distribution over the latent variables $\\dot { \\mathbf { z } } : = ( z _ { i } ) _ { i = 1 } ^ { N }$ . Given an initial value, $\\theta _ { 0 }$ , the EM   \n73 algorithm produces a sequence of estimates, $( \\theta _ { t } ) _ { t = 1 } ^ { T }$ , by alternating between the expectation (E) and   \n74 maximization (M) steps, ",
+        "bbox": [
+            145,
+            273,
+            825,
+            332
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "equation",
+        "img_path": "images/b0f69e94b819f6dec2b2f46df7d892633bd8e60b66ed31dbb1985f882bcb2f60.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\mathrm { E \\mathrm { - } s t e p } ; ~ \\mathcal { Q } _ { t - 1 } ( \\theta ) , } \\\\ & { \\mathrm { M \\mathrm { - } s t e p } ; ~ \\theta _ { t } : = \\arg \\operatorname* { m a x } _ { \\theta \\in \\Theta } \\mathcal { Q } _ { t - 1 } ( \\theta ) . } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            387,
+            337,
+            607,
+            382
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "75 This sequence is guaranteed to monotonically tighten the ELBO, arriving at a local optimum of (2)   \n76 under mild regularity assumptions [56].   \n77 The EM algorithm is computationally expensive, since (4) evaluates $p _ { \\theta } ( z _ { i } , x _ { i } )$ for each $z _ { i } \\in \\mathbb { Z }$ and   \n78 $i \\in ( 1 , \\ldots , N )$ . This has to be performed for all $t \\in ( 1 , \\ldots , T )$ in (5). Albeit the marginal factor,   \n79 $p _ { \\pi _ { z } } ( z )$ , is just the cheap categorical distribution, the conditional factor, $p _ { \\eta _ { z } } ( x | z )$ , typically involves   \n80 high-dimensional operations (e.g., the inversion of the full $D \\times D$ -dimensional covariance matrices   \n81 in the GMMs). Moreover, the M-step (6) is also expensive for large $K$ . This holds despite that (6)   \n82 can be reduced to closed-form updates of expected sufficient statistics for $p _ { \\eta _ { z } } ( x | z )$ belonging to the   \n83 exponential family [44] (again, due to high $D$ ). All in all, the computational complexity of the EM   \n84 algorithm scales with $\\mathcal { O } ( T D N K )$ . ",
+        "bbox": [
+            143,
+            388,
+            825,
+            417
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            145,
+            422,
+            825,
+            535
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "If (6) cannot be computed under a closed-form solution, one can resort to direct gradient-descent optimization of $\\mathcal { Q } ( \\boldsymbol { \\theta } )$ , where arg max is replaced by one (or more) step(s) of a gradient descent technique. The EM algorithm is then referred to as the generalized EM algorithm [56]. ",
+        "bbox": [
+            160,
+            540,
+            823,
+            583
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "88 4 The generalized MHSAEM algorithm ",
+        "text_level": 1,
+        "bbox": [
+            148,
+            602,
+            521,
+            619
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "89 We design a version of the generalized EM algorithm suitable for scenarios where (4) can represent   \n90 deep, discrete, latent variable models, thus being parameterized by possibly complex nonlinear   \n91 transformations. We particularly focus on decreasing the the number of operations in the generalized   \n92 EM algorithm from $\\mathcal { O } ( T D N K )$ to $\\mathcal { O } ( T D B M )$ , where $B \\ll N$ and $M \\ll K$ . ",
+        "bbox": [
+            147,
+            633,
+            825,
+            690
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "4.1 E-step ",
+        "text_level": 1,
+        "bbox": [
+            161,
+            705,
+            256,
+            720
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "94 We reduce the cost of evaluating the EM objective function (4) by combining the minibatching (as   \n95 used many times before) and the Monte Carlo sampling. Namely, the specific application of the latter   \n96 to generic mixture models is the key contribution of this paper.   \n97 Minibatching. At each iteration, $t$ , we compute the conditional expectation in (4) only for a subset—   \n98 here referred to as a minibatch—of the original full dataset, i.e. $( x _ { i } ) _ { i \\in I }$ . Here, $I$ is a set of $B \\ll N$   \n99 indices, $i$ , sampled uniformly without replacement from $( 1 , \\ldots , \\dot { N } )$ . This substantially decreases the   \n100 necessary computations compared to the full sweep over all $N$ datapoints [23].   \n101 Monte Carlo sampling. For each $i \\in I$ , we want to draw $M \\ll K$ random samples from $p _ { \\theta } ( z _ { i } | x _ { i } )$ in   \n102 order to obtain a Monte Carlo estimate of (4). The straightforward way to do this would be to draw   \n103 the samples directly from $p _ { \\theta } ( z _ { i } | x _ { i } )$ . However, direct sampling from $p _ { \\theta } ( z _ { i } | x _ { i } )$ does not lead to any   \n104 substantial decrease in the number of operations. This is caused by the fact that even for a single   \n105 sample of $z _ { i }$ , we have to first compute the normalizing factor, $p _ { \\theta } ( x _ { i } )$ , to obtaining the posterior,   \n106 $p _ { \\theta } ( z _ { i } | x _ { i } )$ . This requires $K$ expensive evaluations of $p _ { \\theta } ( z _ { i } , x _ { i } )$ , which is precisely what we want to   \n107 avoid. Our approach is to resort to the Markov chain Monte Carlo (MCMC), which allows us to   \n108 sample from $p _ { \\theta } ( z _ { i } | x _ { i } )$ , with the computational complexity decreasing to only a single evaluation of   \n109 $p _ { \\theta } ( z _ { i } , x _ { i } )$ per a single sample of $z _ { i }$ .   \n110 MCMC methods obviate the computation of the normalizing factor in $p _ { \\theta } ( z _ { i } | x _ { i } )$ by simulating a   \n111 Markov chain, $( z _ { i , t } ) _ { t = 1 } ^ { T }$ , from a transition kernel, $z _ { i , t } \\sim P ( z _ { i , t - 1 } , \\cdot )$ , which leaves $p _ { \\theta } ( z _ { i } | x _ { i } )$ as its   \n112 unique stationary (invariant) distribution, starting from an initial value $z _ { i , 0 }$ . The specific form of $P$   \n113 determines the structure of an MCMC method. We chose the Metropolis-Hastings (MH) sampler,   \n114 which represents $P ( z _ { i , t - 1 } , z _ { i , t } )$ as follows: given $\\bar { z } _ { i } : = z _ { i , t - 1 }$ , draw a sample from the proposal   \n115 distribution $z _ { i } \\sim q ( \\cdot | \\bar { z } _ { i } )$ , compute the acceptance ratio, ",
+        "bbox": [
+            147,
+            731,
+            825,
+            773
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            147,
+            779,
+            825,
+            835
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            142,
+            842,
+            825,
+            911
+        ],
+        "page_idx": 2
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            140,
+            90,
+            825,
+            147
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            140,
+            152,
+            825,
+            238
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/4e2865ff49fb5c3c34577e0a148cbc695663514dff1302f39bca47fd492f0e3e.jpg",
+        "text": "$$\n\\alpha ( \\bar { z } _ { i } , z _ { i } ) : = \\operatorname* { m i n } \\biggr \\{ 1 , \\frac { p _ { \\eta _ { z _ { i } , t - 1 } } ( x _ { i } | z _ { i } ) \\pi _ { z _ { i } , t - 1 } q ( \\bar { z } _ { i } | z _ { i } ) } { p _ { \\eta _ { \\bar { z } _ { i } , t - 1 } } ( x _ { i } | \\bar { z } _ { i } ) \\pi _ { \\bar { z } _ { i } , t - 1 } q ( z _ { i } | \\bar { z } _ { i } ) } \\biggr \\} ,\n$$",
+        "text_format": "latex",
+        "bbox": [
+            321,
+            241,
+            676,
+            279
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "116 and, if $u < \\alpha \\big ( \\bar { z } _ { i } , z _ { i } \\big )$ —where $u$ is drawn from a uniform distribution, Uniform $( 0 , 1 )$ —accept the   \n117 sample and set $z _ { i , t } = z _ { i }$ ; otherwise, set $z _ { i , t } = \\bar { z } _ { i }$ . For each $i \\in I$ and $t \\in ( 1 , \\ldots , T )$ , we repeat   \n118 this process $M$ times, construing a set $\\mathbf { z } _ { i , t } = ( z _ { i , t } ^ { 1 } , \\dots , z _ { i , t } ^ { M } )$ . Therefore, at every current iteration,   \n119 120 $t$ , we caking $\\bar { z } _ { i } = z _ { i , t - 1 } ^ { M }$ extend the chain from the point where we left at the previous iteration, . Under mild regularity assumptions [52], the chain passes the transiti $t - 1$ , byriod   \n121 (the burn-in phase), and the samples can then be used to approximate the conditional expectation in   \n122 (4) as follows: ",
+        "bbox": [
+            140,
+            282,
+            825,
+            382
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e43005c66ee8eb883ede3aeefb994947aa4f14879ab1b3689fc3176b2f635909.jpg",
+        "text": "$$\n\\widehat { \\mathcal { Q } } _ { t - 1 } ( \\theta ) = \\frac { 1 } { M } \\sum _ { i \\in I } \\sum _ { z \\in \\mathbf { z } _ { i , t } } \\log p _ { \\eta _ { z } } ( x _ { i } | z ) \\pi _ { z } .\n$$",
+        "text_format": "latex",
+        "bbox": [
+            359,
+            378,
+            637,
+            419
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "123 Note that, to ensure this approach is truly efficient, we have to draw only $M \\ll K$ samples at each   \n124 iteration, $t$ ; otherwise, for $M \\approx K$ , we may rather compute the exact marginalization in (4), since it   \n125 is tractable (but computationally costly). ",
+        "bbox": [
+            142,
+            420,
+            825,
+            463
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4.2 M-step ",
+        "text_level": 1,
+        "bbox": [
+            166,
+            478,
+            263,
+            493
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "127 Assume for a moment that (6) with $\\mathcal { Q } _ { t - 1 } ( \\theta )$ given by (8) has a closed-form solution, yielding an   \n128 estimate of $\\theta$ . Such an estimate would have a high variance, converging only for $M \\to \\infty$ and   \n129 $T \\to \\infty$ [19]. The main reason is that the samples would not be reused over the iterations, $t$ ,   \n130 thus wasting computational resources. We consider that there is no closed-form solution of (6),   \n131 and—to ensure that the samples (and thus computations) are recycled over the iterations—we use   \n132 the stochastic approximation (SA) [51] to optimize (8). This is analogous to applying a stochastic   \n133 gradient-descent method, $\\theta _ { t } = \\theta _ { t - 1 } + \\gamma _ { t } \\nabla _ { \\theta } \\tilde { \\mathcal { Q } } _ { t - 1 } ( \\theta )$ , where $\\gamma _ { t }$ is the step-size, satisfying the Robbins  \n134 Monro constraints, $\\begin{array} { r } { \\gamma _ { t } \\in [ 0 , 1 ] , \\sum _ { t \\geq 1 } \\gamma _ { t } = \\infty , \\sum _ { t \\geq 1 } \\gamma _ { t } ^ { 2 } < \\infty , } \\end{array}$ and $\\nabla _ { \\theta }$ is the gradient w.r.t. $\\theta$ . In this   \n135 way, the computations made in $\\nabla _ { \\boldsymbol { \\theta } } \\widehat { \\mathcal { Q } }$ are accumulated via $\\theta _ { t }$ and reused over the iterations.   \n136 The parameters $\\eta _ { z }$ have a different form based on a specific case of $p _ { \\eta _ { z } } ( x | z )$ , whereas $\\pi _ { z }$ is a   \n137 permanent structure in (1). Therefore, without loss of generality, we split (6) into a generic part and a   \n138 fixed part as follows: ",
+        "bbox": [
+            140,
+            502,
+            826,
+            636
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            143,
+            641,
+            828,
+            684
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "equation",
+        "img_path": "images/e1b6543f038daf2078a131f2fa387f7478babb5133579989565df178123b0c85.jpg",
+        "text": "$$\n\\begin{array} { r l } & { \\eta _ { k , t } = \\eta _ { k , t - 1 } + \\gamma _ { t } \\nabla _ { \\eta _ { k } } \\widehat { \\mathcal { Q } } _ { t - 1 } ( \\theta ) , } \\\\ & { \\nu _ { k , t } = \\nu _ { k , t - 1 } + \\gamma _ { t } \\nabla _ { \\nu _ { k } } \\widehat { \\mathcal { Q } } _ { t - 1 } ( \\theta ) , } \\end{array}\n$$",
+        "text_format": "latex",
+        "bbox": [
+            390,
+            688,
+            606,
+            732
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "139 where—to ensure that the probabilities, $( \\pi _ { k , t } ) _ { k = 1 } ^ { K }$ , satisfy the constraints (Section 2)—we transform   \n140 $\\nabla _ { \\pi _ { k } } \\widehat { \\mathcal { Q } }$ via $\\nu _ { k } = \\log \\pi _ { k }$ and optimize w.r.t. $\\nu _ { k }$ . Then, to obtain $( \\pi _ { k , t } ) _ { k = 1 } ^ { K }$ from $\\nu _ { t } : = ( \\nu _ { k , t } ) _ { k = 1 } ^ { K }$ , we   \n141 k b  use the softmax function, i.e. $\\pi _ { k , t } : = \\mathrm { s o f t m a x } ( \\pmb { \\nu } _ { t } ) _ { k } : = \\exp ( \\nu _ { k , t } ) / \\sum _ { l = 1 } ^ { K } \\exp ( \\nu _ { l , t } )$ . ",
+        "bbox": [
+            142,
+            734,
+            825,
+            787
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Computing the gradients for all pairs of $( \\nu _ { k } , \\eta _ { k } ) _ { k = 1 } ^ { K }$ would be inefficient, especially since ${ \\bf z } _ { i , t }$ contains only a small number of unique values of Z for $M \\ll K$ . Consequently, we compute $\\dot { \\nabla } _ { \\eta _ { k } } \\widehat { \\mathcal { Q } }$ and $\\nabla _ { \\nu _ { k } } \\widehat { \\mathcal { Q } }$ only for $k \\in { \\mathrm { u n i q u e } } ( \\mathbf { z } _ { i , t } )$ . We summarize the proposed approach in Algorithm 1. ",
+        "bbox": [
+            160,
+            792,
+            825,
+            843
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "4.3 Proposal distribution ",
+        "text_level": 1,
+        "bbox": [
+            171,
+            857,
+            361,
+            872
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "146 The choice of the proposal distribution has a significant impact on the speed of convergence and the computational cost of the proposed algorithm. Here, we discuss various possible choices of 147 $q \\big ( z _ { i } | \\bar { z } _ { i } \\big )$ . ",
+        "bbox": [
+            145,
+            882,
+            828,
+            912
+        ],
+        "page_idx": 3
+    },
+    {
+        "type": "text",
+        "text": "Input: $\\theta _ { 0 }$ , $( \\mathbf { z } _ { i , 0 } ) _ { i = 1 } ^ { N }$ , $( \\mathbf { x } _ { i } ) _ { i = 1 } ^ { N }$   \nOutput: $( \\theta _ { t } ) _ { t = 1 } ^ { T }$ for $t \\in ( 1 , \\ldots , T )$ or until convergence do form the set $\\dot { I } = ( i _ { j } ) _ { j = 1 } ^ { B }$ by sampling (without replacement) $B$ indices $i \\sim ( 1 , \\dots , N )$ for $i \\in I$ do set $\\bar { z } _ { i }$ as the last element of $\\mathbf { z } _ { i , t - 1 }$ for $j \\in ( 1 , \\ldots , M )$ do sample $z _ { i } \\sim q ( z _ { i } | \\bar { z } _ { i } )$ sample $u \\sim \\mathrm { U n i f o r m } ( 0 , 1 )$ compute $\\alpha ( \\bar { z } _ { i } , z _ { i } )$ in (7) if $u < \\alpha \\big ( \\bar { z } _ { i } , z _ { i } \\big )$ then set $z _ { i , t } ^ { j } = z _ { i }$ and $\\bar { z } _ { i } = z _ { i }$ else set $z _ { i , t } ^ { j } = \\bar { z } _ { i }$ end if end for set $\\mathbf { z } _ { i , t } = ( z _ { i , t } ^ { 1 } , \\dots , z _ { i , t } ^ { M } )$ end for compute (8) compute (9) for $k \\in { \\mathrm { u n i q u e } } ( \\mathbf { z } _ { i , t } )$ compute $\\pi _ { k , t } : = \\mathrm { s o f t m a x } ( \\pmb { \\nu } _ { t } ) _ { k }$ for $k \\in { \\mathord { \\mathbb { Z } } }$ end for ",
+        "bbox": [
+            173,
+            108,
+            787,
+            421
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "148 Optimal proposal $( O )$ . The optimal proposal distribution is $q ( z _ { i } | \\bar { z } _ { i } ) : = q ( z _ { i } ) : = p _ { \\theta } ( z _ { i } | x _ { i } )$ . This   \n149 ensures that the acceptance rate (7) is always $\\alpha ( \\bar { z } _ { i } , z _ { i } ) = 1$ . However, the need to perform $K$   \n150 expensive evaluations of $p _ { \\theta } ( z _ { i } , x _ { i } )$ before sampling from $p _ { \\theta } ( z _ { i } | x _ { i } )$ is the reason we resorted to   \n151 the MH sampler in the first place. We consider this case only to set the upper limit on admissible   \n152 computational cost and to study the impact of sub-optimal proposal distribtions.   \n153 Uniform proposal $( U )$ . The uniform distribution on the discrete interval from 1 to $K$ , i.e. $q ( z _ { i } | \\bar { z } _ { i } ) : =$   \n154 $q ( z _ { i } ) : = \\mathrm { U n i f o r m } ( 1 , K )$ , is the simplest and computationally cheapest variant of the proposal   \n155 distribution. However, due to poor mixing properties, the algorithm may converge slowly for high $K$ .   \n156 Tabular proposal with forgetting $( T F )$ . The key requirement to design a proposal distribution is to   \n157 restrict its computational complexity somewhere between that of the $\\mathrm { U }$ and $\\mathrm { o }$ proposals. One way to   \n158 satisfy this constraint is to use the Markov chain, $( z _ { i , t } ) _ { t = 1 } ^ { T }$ , to learn a transition kernel, $p ( z _ { i } | \\bar { z } _ { i } )$ , see,   \n159 e.g. [3]. Unfortunately, this would require us to store a table with $K ^ { 2 }$ entries for each $i \\in ( 1 , \\ldots , N )$ ,   \n160 161 which is very demanding evethe Markov chain and define: $q ( z _ { i } | \\bar { z } _ { i } ) : = q _ { \\alpha _ { i } } ( z _ { i } ) : = \\mathcal { C } ( \\alpha _ { i } )$ $K$ $N$ herefore, where $\\mathcal { C } ( \\pmb { \\alpha } _ { i } ) \\propto \\Pi _ { k = 1 } ^ { K } \\alpha _ { k , i } ^ { \\bar { 1 ( } z _ { i } = k ) }$ nce inis the   \n162 categorical distribution with the weights $\\pmb { \\alpha } _ { i } : = ( \\alpha _ { 1 , i } , \\dots , \\alpha _ { K , i } )$ . For $\\mathcal { L } ( \\pmb { \\alpha } _ { i } ) : = \\Sigma _ { \\tau = 1 } ^ { t } \\log q _ { \\pmb { \\alpha } _ { i } } ( z _ { i , \\tau } )$   \n163 we obtain an estimate of $\\alpha _ { i }$ at iteration $t$ as follows: $\\begin{array} { r } { \\alpha _ { i , t } : = \\mathrm { \\ a r g m a x } _ { \\alpha _ { i } } \\mathcal L ( \\alpha _ { i } ) \\ = \\ \\frac { n _ { i , t } } { t } } \\end{array}$ t , with   \n164 $n _ { i , t } = \\Sigma _ { \\tau = 1 } ^ { t } \\mathbf { e } _ { z _ { i , t } }$ , where $\\mathbf { e } _ { k }$ is the standard basis vector (a one-hot vector) with one at $k$ th position   \n165 and zeros otherwise. This can be further rewritten into a recursive form: $n _ { i , t } = n _ { i , t - 1 } + \\mathbf { e } _ { z _ { i , t } }$ or,   \n166 using the Robbins-Monro step-size, $n _ { i , t } = ( { \\bf 1 } - { \\bf e } _ { z _ { i , t } } \\gamma _ { t } ) \\odot n _ { i , t - 1 } + \\gamma _ { t } { \\bf e } _ { z _ { i , t } }$ , where 1 is the vector of   \n167 ones, and $\\odot$ is the Hadamard product. We refer to this case simply as “table with forgetting” (TF)   \n168 due to that it represents $N \\times K$ table in the memory and $\\gamma _ { t }$ is a forgetting factor. ",
+        "bbox": [
+            140,
+            449,
+            825,
+            518
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            142,
+            525,
+            826,
+            568
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            140,
+            573,
+            826,
+            763
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "169 5 Related work ",
+        "text_level": 1,
+        "bbox": [
+            145,
+            782,
+            316,
+            799
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "text",
+        "text": "Stochastic approximation expectation-maximization. The application of SA to prevent the evaluation   \n1 of all $K$ components in mixture models has been overlooked for a long time. The reason is that the   \n72 original motivation to combine the EM algorithm with SA is to address the analytical intractability   \n73 of the expected value under $p _ { \\theta } ( z | x )$ in (4), which is, however, almost always tractable for mixture   \n74 models. The intractability issue is addressed by either the Monte Carlo SAEM (MCSAEM) [12]   \n75 or the Markov chain Monte Carlo SAEM (MCMCSAEM) [31]. Applying the former approach to   \n76 mixture models would be inefficient, since it evaluates $K$ joint distributions, $p _ { \\theta } ( z , x )$ , before drawing   \n177 $M$ samples from $p _ { \\theta } ( z | x )$ . Therefore, this method reduces only the computational cost of updating the   \n178 sufficient statistics. This is addressed by the latter approach, where $M < K$ samples from a proposal   \n179 distribution, $q ( z | x )$ , is used to calculate $p _ { \\theta } ( z , x )$ and also the sufficient statistics. However, all these   \n180 methods process all data at every iteration, providing only a limited advantage over the conventional   \n181 EM algorithm. Minibatch versions of these techniques have recently been proposed [27, 32, 1].   \n182 All the above methods commonly assume $p _ { \\theta } ( z , x )$ belonging to the exponential family. This provides   \n183 a convenient, but limiting, property which allows (6) to be computed under a closed-form solution.   \n184 The main contribution of our work is to release this restrictive assumption by admitting that $p _ { \\theta } ( z , x )$   \n185 (and thus $\\mathcal { Q }$ ) is given by possibly complex and intractable transformations.   \n186 Sparse and truncated variational techniques. There is only a small body of methods explicitly   \n187 reducing the number of evaluated components. Their common aspect is that they follow from the   \n188 variational framework, where the exact posterior, $p _ { \\theta } ( z | x )$ , is approximated by a variational posterior,   \n189 $q ( z | x )$ . This sparse, approximate, posterior is defined over a lower number of components, $M \\ll K$ ,   \n190 such that only the important components are selected, relying on relaxation of the hard EM algorithm   \n191 from taking a single $M = 1$ assignment [26] to taking multiple $M \\ll K$ assignments. The sparse   \n192 SAEM (SSAEM) algorithm [24] selects the components by a quick partial sorting of the posterior   \n193 probabilities, $p _ { \\theta } ( z | x )$ . Again, this requires $K$ evaluations of $p _ { \\theta } ( z , x )$ before the sorting, thus only   \n194 reducing the amount of updated statistics. Similarly, the truncated SAEM (TSAEM) algorithm [18]   \n195 selects $M < K$ cluster-to-cluster and $\\bar { M } < K$ cluster-to-datapoint minimal Euclidean distances,   \n196 preventing the problem in the SSAEM algorithm. However, all these distances are evaluated for all   \n197 components in a pairwise manner, leading to $K ^ { 2 }$ -computational complexity, which makes the saving   \n198 dubious. Similarly as before, these methods assume $p _ { \\theta } ( z , x )$ to belong to the exponential family. ",
+        "bbox": [
+            155,
+            814,
+            823,
+            911
+        ],
+        "page_idx": 4
+    },
+    {
+        "type": "image",
+        "img_path": "images/0c516376ee7ecf27c7edaec6674fb19224d829bc10798ac37d421adee064aa0e.jpg",
+        "image_caption": [
+            "Figure 1: The training log-likelihood, $\\mathcal { L } ( \\boldsymbol { \\theta } _ { t } )$ , versus the computational time (in seconds). Here, on the $\\mathbf { X }$ -axis, the computational time at a current iteration, $t$ , is obtained by accumulating the time from the previous iterations. corresponds to $\\mathcal { L } ( \\boldsymbol { \\theta } _ { t _ { 9 5 } } )$ , where $t _ { 9 5 }$ is the iteration of reaching $9 5 \\%$ of max $\\mathcal { L } ( \\boldsymbol { \\theta } _ { t } )$ . The projection of $^ { \\circ }$ on the $\\mathbf { X }$ -axis gives the time to reach $\\mathcal { L } ( \\boldsymbol { \\theta } _ { t _ { 9 5 } } )$ . This experiment was performed with the following settings: $( D , \\bar { K } , N , \\omega , B , M , T ) = ( 1 0 , 1 0 0 , 1 0 k , 0 . 1 , \\bar { 2 } 0 0 , 2 , 2 0 k )$ , see Section 6.1 for details. The results are averaged over five repetitions. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            285,
+            92,
+            699,
+            236
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            142,
+            356,
+            825,
+            426
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            142,
+            433,
+            825,
+            488
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            140,
+            494,
+            826,
+            674
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "199 We summarize the distinguishing features of the above discussed methods in Table 1. ",
+        "bbox": [
+            151,
+            680,
+            728,
+            695
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "6 Experiments ",
+        "text_level": 1,
+        "bbox": [
+            166,
+            714,
+            312,
+            731
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "To demonstrate the key features of our algorithm—its low computational complexity, competitive learning performance, and generality—we use it below to train: (i) GMMs on synthetic datasets, and (ii) SPTNs [47] and (iii) mixtures of real NVP flows [48] on real datasets. All experiments have been performed on a Slurm cluster equipped with Intel Xeon Scalable Gold 6146 with 384GB of RAM. ",
+        "bbox": [
+            174,
+            744,
+            825,
+            800
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "6.1 Gaussian mixture models ",
+        "text_level": 1,
+        "bbox": [
+            171,
+            816,
+            388,
+            830
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "text",
+        "text": "Consider the special case of a data-generating distribution given by (1), with the components taking the form of the multivariate Gaussian distribution, $p _ { \\eta _ { z } } ( x | z ) = \\mathcal { N } ( x ; \\mu _ { z } , \\Sigma _ { z } )$ , where $\\mu _ { z }$ is the mean value and $\\Sigma _ { z }$ is the covariance matrix. The difficulty of learning GMMs heavily depends on the degree of interaction among all mixture components, hence having the ability to generate synthetic datasets with arbitrary overlap characteristics between all pairs of components is crucial for systematic ",
+        "bbox": [
+            173,
+            842,
+            823,
+            911
+        ],
+        "page_idx": 5
+    },
+    {
+        "type": "image",
+        "img_path": "images/0ced572977ae404a4e785379dc4987ce758b6f6930ca4d73adbe35955ed678b7.jpg",
+        "image_caption": [
+            "Figure 2: The absolute error, $\\mathrm { A E } = | \\mathcal { L } ( \\theta _ { t _ { 9 5 } } ) - \\mathcal { L } ( \\theta ) |$ , versus the computational time (in seconds). All experiments use the following settings: $( D , K , \\dot { N } , \\omega , B , M , T ) = \\bar { ( } 1 0 , 1 0 0 , 1 0 k , 0 . 1 , 2 0 0 , 2 , 2 0 k )$ , where the number of components, $K$ , (left), the batchsize, $B$ , (middle) and the number of samples, $M$ , (right) change for different values denoted by $( + , \\sqsupset , \\circ , \\pmb { \\triangle } )$ . At each of these points (marks), we perform an experiment as illustrated in Figure 1, find $\\mathcal { L } ( \\boldsymbol { \\theta } _ { t _ { 9 5 } } )$ to compute the AE, and record the time corresponding to $t _ { 9 5 }$ . The results are averaged over five repetitions. "
+        ],
+        "image_footnote": [],
+        "bbox": [
+            196,
+            89,
+            805,
+            250
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "211 evaluation of performance of learning algorithms [43]. Traditional techniques usually define overlap   \n212 (or separation) of components only in terms of their mean vectors and maximum eigenvalues of the   \n213 covariance matrices, not accounting for their rotation and mixing weights (see [36] for a detailed   \n214 treatment of the problem). We therefore use a more objective measure of the clustering complexity   \n215 defined by the total probability of misclassification [41], which allows to generate data with a   \n216 user-defined degree of maximum pairwise overlap, $\\omega$ .   \n217 Experiment settings: We generate the parameters of (1), and the corresponding dataset, uniquely for a   \n218 given quadruple $( D , K , N , \\omega )$ . Therefore, the parameters of the generative model are known and we   \n219 can measure and display the convergence of the training log-likelihood, $\\mathcal { L } ( \\boldsymbol { \\theta } _ { t } )$ , compared to the exact   \n220 log-likelihood, $\\mathcal { L } ( \\boldsymbol { \\theta } ) \\dot { }$ , for $t = ( 1 , \\ldots , T )$ . We are further interested in the absolute error between the   \n221 training log-likelihood at the iteration of reaching $9 5 \\%$ of its maximum value, $t _ { 9 5 }$ , and the exact   \n222 log-likelihood, i.e. $\\mathrm { A E } = | \\mathcal { L } ( \\theta _ { t _ { 9 5 } } ) - \\mathcal { L } ( \\theta ) |$ .   \n23 We also measure the computational time until reaching $t _ { 9 5 }$ . We have used $9 5 \\%$ of the maximum   \n24 value instead of the maximum value to prevent cases, where the model oscillate around target value,   \n25 making the estimate of convergence time very noisy (for example MCSAEM in Figure 1).   \n226 Algorithms: The GMMs belong to the exponential family of probability distributions. This allows us   \n227 to find a closed-form, recursive, solution of (6), relying on a Robbins-Monro type of the step-size   \n228 sequence, $( \\gamma _ { t } ) _ { t = 1 } ^ { T }$ , [7, 44]. In this setting, we compare our MHSAEM algorithm with a number of   \n229 related methods in Table 1. Note we use the acronyms U and TF to specify the proposal distribution of   \n230 the MHSAEM algorithm (Section 4.3). However, we do not use the O-proposal, since the MHSAEM  \n231 O algorithm is equivalent to the MCSAEM algorithm. All the SA-variants in Table 1 use a minibatch   \n232 of size $B$ . The key quantity to reduce the number of evaluated components and/or sufficient statistics   \n233 in the SSAEM, TSAEM, MCSAEM and MHSAEM algorithms is collectively denoted by $M$ (Section   \n234 5). Note that we always keep $M = \\bar { M }$ in the TSAEM algorithm (see Figure 1 and 2 for concrete   \n235 numbers). We use the step-size given by $\\gamma _ { t } = 1$ for $t = 1 , \\ldots , 5 0$ and $\\gamma _ { t } = 0 . 0 5$ otherwise. In   \n236 this section, to counteract the issue of attaining poor local optima, we equip all algorithms with the   \n237 anti-annealing schedule $( \\beta _ { t } ) _ { t = 1 } ^ { T }$ , starting with $\\beta _ { 1 } = 0 . 1$ , reaching $\\beta _ { 2 / 3 T } = 1 . 2$ , and decreasing back   \n238 to $\\beta _ { T } = 1 . 0$ , see [43] for details. The initial estimates of: (i) $\\mu _ { k }$ are uniformly drawn from the unit   \n239 hyper-cube, (ii) $\\Sigma _ { k }$ are fixed to unit diagonal matrix, and (iii) $\\pi _ { k }$ are uniformly drawn from the unit   \n240 interval (followed by normalization).   \n241 Results: Figure 1 shows that the EM [13] and SAEM [44] algorithms take the longest time to   \n242 converge, attaining a poor local optima. On the other hand, the MCSAEM [1] and MHSAEM (U   \n243 and TF) algorithms achieve $\\mathcal { L } ( \\boldsymbol { \\theta } _ { t _ { 9 5 } } )$ closest to the likelihood $\\mathcal { L } ( \\boldsymbol { \\theta } )$ of the true model. Moreover, both   \n244 MHSAEM algorithms reach this value in the shortest time compared to all the other methods. The   \n245 SSAEM [24] and TSAEM [18] algorithms are comparable in terms of the computational time, but   \n246 they both provide the lowest $\\mathcal { L } ( \\boldsymbol { \\theta } _ { t _ { 9 5 } } )$ . In Figure 2, we investigate sensitivity of fitting the model to   \n47 increasing values of $K$ , $B$ and $M$ by measuring the time and the likelihood again. In all the cases,   \n48 the proposed MHSAEM algorithms achieve the lowest AE in the shortest time. ",
+        "bbox": [
+            140,
+            387,
+            825,
+            470
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            140,
+            477,
+            825,
+            561
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            156,
+            566,
+            826,
+            608
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            140,
+            614,
+            826,
+            821
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            140,
+            828,
+            825,
+            912
+        ],
+        "page_idx": 6
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            151,
+            90,
+            825,
+            119
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "SSAEM and TSAEM algorithms failed to converge for $M > 2$ and for $K > 5 0$ respectively. We believe this is caused by selecting only $M$ maximal probabilities in the SSAEM (or distances in the TSAEM) algorithm (Section 5), which prevents certain, but not a negligible number of, components from being updated, thus providing only a crude approximation of $\\bar { p } _ { \\theta } \\bar { ( } z | x )$ . The results then suffer from substantial variational gap to the exact log-likelihood (Figure 1). On the contrary, MH sampler provides samples which consistently approximate $p _ { \\theta } ( z | x )$ despite evaluating much lower number of components in each step. ",
+        "bbox": [
+            173,
+            126,
+            825,
+            223
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "6.2 Sum-product transform networks ",
+        "text_level": 1,
+        "bbox": [
+            166,
+            239,
+            446,
+            253
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "The sum product networks (SPNs) are a deep learning extension of finite mixture models. They can be interpreted as a mixture of trees [60], where each tree corresponds to a component. Therefore, they can be cast into the form of (1), but the number of components grows exponentially with their depth. In this section, we use recently proposed SPTNs which introduce additional transformation nodes to provide better expressiveness than the SPNs (SPTNs effectively generalize SPNs and flow models into one large family of models). ",
+        "bbox": [
+            173,
+            265,
+            825,
+            348
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Experimental settings: We use 19 real datasets from the UCI database [16, 37, 35, 54], preprocessed in the same way as in [46]. For each experiment, we randomly split the data into $64 \\%$ , $16 \\%$ and $20 \\%$ for training, validation and testing, respectively. We calculate the average log-likelihood on the test set and measure again the time to reach $9 5 \\%$ of the maximal training log-likelihood, $\\mathcal { L } ( \\boldsymbol { \\theta } _ { t _ { 9 5 } } )$ . ",
+        "bbox": [
+            171,
+            354,
+            825,
+            410
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "To evaluate various (possibly shallow and/or deep) architectures of SPTNs, we fit each dataset with all the following combinations of hyper-parameters1: $s \\in ( 8 , 3 2 , 1 2 8 )$ , $b \\in ( 2 , 4 , 6 , 8 )$ , $l \\in ( 2 , 3 , 4 )$ , where $s$ is the number of children of each sum node, $b$ is the number of partitions of each product node, and $l$ is the number of layers (one layer contains sum and product nodes). The number of components of the SPTN, after its conversion into (1), is given as follows: $K = s ^ { l }$ . Note that the maximum number of components for the investigated parameters of the SPTN is 268,435,456. To reduce the space of possible architectures, we restrict ourselves only to (i) the leaf nodes given by $\\mathcal { N } ( 0 , \\bf { I } )$ ; (ii) affine transformations fixed to the singular value decomposition, choosing the the Givens parameterization for the unitary matrices [47]; and (iii) no sharing of any type of nodes [47]. ",
+        "bbox": [
+            161,
+            416,
+            825,
+            541
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "276 Algorithms: We evaluate only on the MHSAEM-U algorithm—due to its favourable computational   \n277 complexity and simplicity—and compare it with the stochastic gradient-descent (SGD) algorithm,   \n278 which is routinely used to train SP(T)Ns [45, 47]. In this case, SGD in each iteration performs   \n279 computations over all subtrees of the network, whereas the MHSAEM-U algorithm computes with   \n280 only $M = 1$ subtrees, thus we should observe speed-up of the computations. In our implementation,   \n281 both these methods perform optimization of their respective objective functions—the log-likelihood   \n282 (2) for SGD and the EM objective (8) for MHSAEM-U—via the use of the automatic differentiation   \n283 and the ADAM optimizer [29], using $B = 1 0 0$ and $T = 2 0 0 0 0$ . ",
+        "bbox": [
+            142,
+            547,
+            825,
+            659
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "Results: Since each dataset might benefit from a different architecture, Table 6.2 shows the test log-likelihood of the architectures selected according to the best likelihood measured on the validation set and the corresponding speed-up. The test log-likelihoods reveal that the MHSAEM-U algorithm outperforms the SGD algorithm on 10 out of 19 datasets, which was not originally the goal, but the added stochasticity helps to escape poor local minima. The speed-up demonstrates lower computational complexity of the MHSAEM-U algorithm on 17 out of 19 datasets, which was the main goal. The magic-telescope and wine datasets show approximately $1 0 2 \\times$ and $7 5 \\times$ speed-up, respectively, while on very small datasets (pima-indians and iris), the SGD is faster due to effective implementation. In the supplementary material, we present Table 3, exhibiting the same trends on a fixed architecture. ",
+        "bbox": [
+            171,
+            665,
+            825,
+            803
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "6.3 Mixtures of real NVP flows ",
+        "text_level": 1,
+        "bbox": [
+            173,
+            819,
+            400,
+            834
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "text",
+        "text": "We consider another class of mixture models (1), where each component $p _ { \\eta _ { z } } ( x | z )$ is transformed by the flow model—real NVP [15]. These transformations are parameterized via deep neural networks, allowing for flexible adjustment of the learning capacity of each component. ",
+        "bbox": [
+            176,
+            845,
+            825,
+            887
+        ],
+        "page_idx": 7
+    },
+    {
+        "type": "table",
+        "img_path": "images/82dc83ba06aea51dfe211f3df1e7f4eb07c99ebac951998df4d56e6c0d99003a.jpg",
+        "table_caption": [
+            "Table 2: The speed-up and test log-likelihood, $\\mathcal { L } ^ { \\mathrm { t e s t } }$ , for the SGD and MHSAEM-U algorithms. The test log-likelihood (higher is better) is computed for the best model, with the corresponding $K$ , which is selected based on the validation log-likelihood. The speed-up is computed as the ratio of MHSAEM-U to SGD, i.e. their time to reach $9 5 \\%$ of the training log-likelihood. The results are averaged over five repetitions. Then, the higher test log-likelihood is highlighted with bold blue, and and no speed-up is highlighted with red. The average rank is computed as the standard competition (“1224”) ranking [14] on each dataset (lower is better). "
+        ],
+        "table_footnote": [],
+        "table_body": "<table><tr><td rowspan=\"3\"></td><td colspan=\"5\">Sum-product transformnetworks</td><td colspan=\"5\">Mixtures of real NVP flows SGD</td></tr><tr><td rowspan=\"2\"></td><td colspan=\"2\">SGD</td><td colspan=\"2\">MHSAEM-U</td><td colspan=\"2\"></td><td colspan=\"2\"></td><td colspan=\"2\">MHSAEM-U</td></tr><tr><td>speed-up</td><td>Ltest</td><td>K</td><td>Ltest</td><td>K</td><td>speed-up</td><td>Ltest</td><td>K 32</td><td>Ltest</td><td>K</td></tr><tr><td>breast-cancer-wisconsin</td><td>4.66</td><td>-4.66</td><td>64</td><td>1.43</td><td>1024</td><td>0.63</td><td>-99.85</td><td></td><td>-39.31</td><td></td><td>128</td></tr><tr><td>cardiotocography</td><td>10.55</td><td>59.52</td><td>512</td><td>31.04</td><td>1024</td><td></td><td>9.85</td><td>54.34</td><td>32</td><td>56.08</td><td>128</td></tr><tr><td>magic-telescope</td><td>102.53</td><td>-3.65</td><td>512</td><td>-5.03</td><td>1024</td><td></td><td>3.74</td><td>-3.97</td><td>8</td><td>-4.22</td><td>8</td></tr><tr><td>pendigits</td><td>4.89</td><td>0.88</td><td>1024</td><td>-4.86</td><td>16384</td><td></td><td>4.17</td><td>1.46</td><td>8</td><td>0.48</td><td>8</td></tr><tr><td>pima-indians</td><td>0.37</td><td>-8.54</td><td>64</td><td>-7.62</td><td></td><td>64</td><td>1.35</td><td>-20.09</td><td>128</td><td>-16.33</td><td>128</td></tr><tr><td>wall-following-robot</td><td>3.43</td><td>1.84</td><td>1024</td><td>-11.3</td><td>16384</td><td></td><td>22.21</td><td>-14.26</td><td>128</td><td>-17.56</td><td>128</td></tr><tr><td>waveform-1</td><td>4.35</td><td>-26.14</td><td>64</td><td>-23.91</td><td>1024</td><td></td><td>3.72</td><td>-34.12</td><td>8</td><td>-33.42</td><td>8</td></tr><tr><td>waveform-2</td><td>4.82</td><td>-26.21</td><td>64</td><td>-23.91</td><td></td><td>1024</td><td>4.12</td><td>-34.15</td><td>8</td><td>-33.64</td><td>8</td></tr><tr><td>yeast</td><td>20.57</td><td>10.26</td><td>512</td><td>5.18</td><td>1024</td><td></td><td>14.49</td><td>6.61</td><td>128</td><td>9.59</td><td>128</td></tr><tr><td>ecoli</td><td>1.86</td><td>-5.5</td><td>64</td><td>-0.22</td><td>1024</td><td></td><td>2.15</td><td>-11.37</td><td>128</td><td>-10.64</td><td>128</td></tr><tr><td>ionosphere</td><td>1.88</td><td>-20.27</td><td>64</td><td>-5.93</td><td></td><td>512</td><td>2.74</td><td>-87.01</td><td>128</td><td>-42.75</td><td>128</td></tr><tr><td>iris</td><td>0.23</td><td>-10.65</td><td>64</td><td>-1.49</td><td>16384</td><td></td><td>3.28</td><td>-16.34</td><td>128</td><td>-9.21</td><td>32</td></tr><tr><td>page-blocks</td><td>12.18</td><td>12.21</td><td>512</td><td>6.84</td><td>1024</td><td></td><td>44.95</td><td>17.13</td><td>128</td><td>17.94</td><td>32</td></tr><tr><td>parkinsons</td><td>1.46</td><td>-21.85</td><td>64</td><td>0.5</td><td></td><td>512</td><td>3.09</td><td>-566.58</td><td>128</td><td>-33.31</td><td>32</td></tr><tr><td>sonar</td><td>2.96</td><td>-95.39</td><td>512</td><td>-69.29</td><td></td><td>64</td><td>2.52</td><td>-622.2</td><td>128</td><td>-88.81</td><td>128</td></tr><tr><td>statlog-segment</td><td>1.44</td><td>47.35</td><td>512</td><td>26.53</td><td>16384</td><td></td><td>38.49</td><td>35.84</td><td>128</td><td>42.04</td><td>32</td></tr><tr><td>statlog-vehicle</td><td>2.97</td><td>-4.25</td><td>64</td><td>-5.45</td><td>1024</td><td></td><td>6.78</td><td>-31.34</td><td>32</td><td>-26.43</td><td>128</td></tr><tr><td>wine rank</td><td>75.42</td><td>-25.99</td><td>1024</td><td>-13.27</td><td></td><td>1024</td><td>2.05</td><td>-171.58</td><td>128</td><td>-25.57</td><td>128</td></tr><tr><td></td><td></td><td>1.56</td><td></td><td>1.44</td><td></td><td></td><td></td><td>1.83</td><td></td><td>1.17</td><td></td></tr></table>",
+        "bbox": [
+            186,
+            194,
+            813,
+            422
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "298 Experimental settings: We use the same experimental settings and evaluation metrics as in Section   \n299 6.2. We apply the mixture model on all datasets, changing the number of components as follows:   \n300 $K \\in ( 8 , 3 2 , 1 2 8 )$ . Each real NVP-based component in the mixture model has (i) the translation   \n301 function parameterized via multi-layer perceptron with a single hidden layer of dimension 10, using   \n302 the rectified linear activation function; and (ii) the scale function parameterized via the same network   \n303 except with the hyperbolic tangent activation function. We do not use the batch normalization [15] and   \n304 we stack two layers of the translation-scale transformation (we have used implementation from [20]). ",
+        "bbox": [
+            147,
+            455,
+            825,
+            553
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "Algorithms: The algorithms and their settings are the same as those in Section 6.2. ",
+        "bbox": [
+            161,
+            559,
+            710,
+            574
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "306 Results: The experimental results are presented in right part of Table 6.2. They are similar to those   \n307 obtained in the previous section. In terms of the test log-likelihood, the MHSAEM-U algorithm   \n308 outperforms the SGD algorithm on all but three datasets, and it provides a substantial speed-up on all   \n309 datasets except one. The test likelihood of models with the real NVP flows is most of the time worse   \n310 than that of SPTNs with the affine transformations. As explained in the supplementary, this is due to   \n311 the overfitting, which has been observed in [47]. ",
+        "bbox": [
+            140,
+            580,
+            825,
+            664
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "312 7 Conclusion ",
+        "text_level": 1,
+        "bbox": [
+            142,
+            689,
+            299,
+            707
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "313 This paper has presented a method to decrease computational complexity of fitting mixture models,   \n314 including their generalizations, such as sum-product-(transform) networks and mixtures of flow   \n315 models. The speed-up is achieved by evaluating and updating only a single component (per iteration),   \n316 where the Metropolis-Hasting algorithm ensures sampling of components from a proper posterior. An   \n317 experimental comparison on all three classes of models mentioned above confirmed the theoretical   \n318 expectations. The method significantly speeds-up the fitting time and, importantly, without sacrificing   \n319 the quality of the fit. In fact, the likelihood was better than that of the models fitted by the EM   \n320 algorithm or the SGD algorithm in more than $50 \\%$ of cases. We attribute this to higher stochasticity,   \n321 which helps to escape from poor local minima.   \n322 In the experiments, the proposed method has used a uniform proposal distribution in the MH sampler.   \n323 Despite outperforming the alternative methods, we conjecture that this limits the speed of convergence.   \n324 Therefore, we believe that there is still a room for improvement in the implementation. We plan to   \n325 address these issues in future work. ",
+        "bbox": [
+            140,
+            724,
+            825,
+            849
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "",
+        "bbox": [
+            142,
+            856,
+            825,
+            911
+        ],
+        "page_idx": 8
+    },
+    {
+        "type": "text",
+        "text": "The presented method decreases the computational complexity of fitting large (and deep) mixture models, which leads to five to hundred time speed-up depending on a size of the problem (although negative exceptions occurs). We believe this line of research, which we want to continue, to have important benefits. First, it is directly related to decrease in energy consumption and in production of CO2 (we expect similar rates as the speedup). Second, it has a positive effect on financial aspects of deploying (and experimenting with) mixture models. Third, it decreases the hardware requirements, as in all experiments presented above the model was fitted on a single-core. ",
+        "bbox": [
+            173,
+            121,
+            826,
+            218
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "References ",
+        "text_level": 1,
+        "bbox": [
+            169,
+            239,
+            266,
+            255
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "[1] S. Allassonnière and J. Chevallier. A new class of stochastic EM algorithms: Escaping local maxima and handling intractable sampling. Computational Statistics & Data Analysis, 159:107159, 2021.   \n[2] S. Ayesha, M. K. Hanif, and R. Talib. Overview and comparative study of dimensionality reduction techniques for high dimensional data. Information Fusion, 59:44–58, 2020.   \n[3] D. S. Bai. Efficient estimation of transition probabilities in a Markov chain. The Annals of Statistics, pages 1305–1317, 1975.   \n[4] D. M. Blei, A. Kucukelbir, and J. D. McAuliffe. Variational inference: A review for statisticians. Journal of the American statistical Association, 112(518):859–877, 2017.   \n[5] J. Blömer and K. Bujna. Adaptive seeding for Gaussian mixture models. In Pacific-asia conference on knowledge discovery and data mining, pages 296–308. Springer, 2016.   \n[6] T. Campbell and B. Beronov. Sparse variational inference: Bayesian coresets from scratch. arXiv preprint arXiv:1906.03329, 2019.   \n[7] O. Cappé and E. Moulines. On-line expectation–maximization algorithm for latent data models. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 71(3):593–613, 2009.   \n[8] J. Chen, J. Zhu, Y. Teh, and T. Zhang. Stochastic expectation maximization with variance reduction. Advances in Neural Information Processing Systems, page 7967, 2018.   \n[9] N. Chopin. A sequential particle filter method for static models. Biometrika, 89(3):539–552, 2002.   \n[10] G. Compiani and Y. Kitamura. Using mixtures in econometric models: A brief review and some new results. The Econometrics Journal, 19(3):C95–C127, 2016.   \n[11] S. Dasgupta. Learning mixtures of Gaussians. In 40th Annual Symposium on Foundations of Computer Science (Cat. No. 99CB37039), pages 634–644. IEEE, 1999.   \n[12] B. Delyon, M. Lavielle, E. Moulines, et al. Convergence of a stochastic approximation version of the EM algorithm. The Annals of Statistics, 27(1):94–128, 1999.   \n[13] A. P. Dempster, N. M. Laird, and D. B. Rubin. Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society: Series B (Methodological), 39(1):1–22, 1977.   \n[14] J. Demšar. Statistical comparisons of classifiers over multiple data sets. The Journal of Machine Learning Research, 7:1–30, 2006.   \n[15] L. Dinh, J. Sohl-Dickstein, and S. Bengio. Density estimation using real NVP. In 5th International Conference on Learning Representations, ICLR 2017, 2017.   \n[16] D. Dua and C. Graff. UCI machine learning repository, 2017. ",
+        "bbox": [
+            178,
+            260,
+            828,
+            917
+        ],
+        "page_idx": 9
+    },
+    {
+        "type": "text",
+        "text": "[17] D. Feldman, M. Faulkner, and A. Krause. Scalable training of mixture models via coresets. In Proceedings of the 24th International Conference on Neural Information Processing Systems, pages 2142–2150, 2011. 372 [18] D. Forster and J. Lücke. Can clustering scale sublinearly with its clusters? A variational EM acceleration of GMMs and k-means. In International Conference on Artificial Intelligence and Statistics, pages 124–132. PMLR, 2018. [19] G. Fort, E. Moulines, et al. Convergence of the Monte Carlo expectation maximization for curved exponential families. Annals of Statistics, 31(4):1220–1259, 2003. [20] J. Franc˚u. Continuousflows.jl. https://github.com/janfrancu/ContinuousFlows.jl, 2020. [21] K. Greff, S. van Steenkiste, and J. Schmidhuber. Neural expectation maximization. In Proceedings of the 31st International Conference on Neural Information Processing Systems, pages 6694–6704, 2017. [22] R. Gribonval, A. Chatalic, N. Keriven, V. Schellekens, L. Jacques, and P. Schniter. Sketching datasets for large-scale learning (long version). arXiv preprint arXiv:2008.01839, 2020. [23] M. D. Hoffman, D. M. Blei, C. Wang, and J. Paisley. Stochastic variational inference. Journal of Machine Learning Research, 14(5), 2013. [24] M. C. Hughes and E. B. Sudderth. Fast learning of clusters and topics via sparse posteriors. arXiv preprint arXiv:1609.07521, 2016. [25] K. Humphreys and D. Titterington. Approximate Bayesian inference for simple mixtures. In COMPSTAT, pages 331–336. Springer, 2000. [26] B.-H. Juang and L. R. Rabiner. The segmental K-means algorithm for estimating parameters of hidden Markov models. IEEE Transactions on acoustics, speech, and signal Processing, 38(9):1639–1641, 1990. [27] B. Karimi, M. Lavielle, and É. Moulines. On the convergence properties of the mini-batch EM and MCEM algorithms, 2019. [28] N. Keriven, A. Bourrier, R. Gribonval, and P. Pérez. Sketching for large-scale learning of mixture models. Information and Inference: A Journal of the IMA, 7(3):447–508, 2018. [29] D. P. Kingma and J. Ba. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014. [30] A. Kucukelbir, D. Tran, R. Ranganath, A. Gelman, and D. M. Blei. Automatic differentiation variational inference. The Journal of Machine Learning Research, 18(1):430–474, 2017. [31] E. Kuhn and M. Lavielle. Coupling a stochastic approximation version of EM with an MCMC procedure. ESAIM: Probability and Statistics, 8:115–131, 2004. [32] E. Kuhn, C. Matias, and T. Rebafka. Properties of the stochastic approximation EM algorithm with mini-batch sampling. Statistics and Computing, 30(6):1725–1739, 2020. [33] A. Lagrange, M. Fauvel, and M. Grizonnet. Large-scale feature selection with Gaussian mixture models for the classification of high dimensional remote sensing images. IEEE Transactions on Computational Imaging, 3(2):230–242, 2017. [34] M. Lavine and M. West. A Bayesian method for classification and discrimination. Canadian Journal of Statistics, 20(4):451–461, 1992. [35] M. Little, P. McSharry, S. Roberts, D. Costello, and I. Moroz. Exploiting nonlinear recurrence and fractal scaling properties for voice disorder detection. Nature Precedings, pages 1–1, 2007. [36] R. Maitra and V. Melnykov. Simulating data to study performance of finite mixture modeling and clustering algorithms. Journal of Computational and Graphical Statistics, 19(2):354–376, 2010. ",
+        "bbox": [
+            145,
+            93,
+            828,
+            920
+        ],
+        "page_idx": 10
+    },
+    {
+        "type": "text",
+        "text": "15 [37] O. L. Mangasarian and W. H. Wolberg. Cancer diagnosis via linear programming. Technical report, University of Wisconsin-Madison Department of Computer Sciences, 1990.   \n17 [38] J.-M. Marin, K. Mengersen, and C. P. Robert. Bayesian modelling and inference on mixtures of distributions. Handbook of statistics, 25:459–507, 2005.   \n19 [39] C. A. McGrory, D. C. Ahfock, J. A. Horsley, and C. L. Alston. Weighted Gibbs sampling for mixture modelling of massive datasets via coresets. Stat, 3(1):291–299, 2014. [40] G. J. McLachlan, S. X. Lee, and S. I. Rathnayake. Finite mixture models. Annual review of statistics and its application, 6:355–378, 2019.   \n423 [41] V. Melnykov, W.-C. Chen, and R. Maitra. MixSim: An R package for simulating data to study performance of clustering algorithms. Journal of Statistical Software, 51(12):1, 2012.   \n25 [42] T. Monnier, T. Groueix, and M. Aubry. Deep transformation-invariant clustering. In Conference on Neural Information Processing Systems (NeurIPS 2020), 2020. [43] I. Naim and D. Gildea. Convergence of the EM algorithm for Gaussian mixtures with unbalanced mixing coefficients. In Proceedings of the 29th International Coference on International Conference on Machine Learning, pages 1427–1431, 2012. [44] H. D. Nguyen, F. Forbes, and G. J. McLachlan. Mini-batch learning of exponential family finite mixture models. Statistics and Computing, pages 1–18, 2020. [45] R. Peharz, A. Vergari, K. Stelzner, A. Molina, X. Shao, M. Trapp, K. Kersting, and Z. Ghahramani. Random sum-product networks: A simple and effective approach to probabilistic deep learning. In Uncertainty in Artificial Intelligence, pages 334–344. PMLR, 2020. [46] T. Pevný. Loda: Lightweight on-line detector of anomalies. Machine Learning, 102(2):275–304, 2016. [47] T. Pevný, V. Šmídl, M. Trapp, O. Polácek, and T. Oberhuber. Sum-product-transform networks: ˇ Exploiting symmetries using invertible transformations. arXiv preprint arXiv:2005.01297, 2020. [48] G. G. Pires and M. A. Figueiredo. Variational mixture of normalizing flows. arXiv preprint arXiv:2009.00585, 2020.   \n42 [49] A. Rau, C. Maugis-Rabusseau, M.-L. Martin-Magniette, and G. Celeux. Co-expression analysis of high-throughput transcriptome sequencing data with Poisson mixture models. Bioinformatics, 31(9):1420–1427, 2015.   \n45 [50] R. A. Redner and H. F. Walker. Mixture densities, maximum likelihood and the EM algorithm. SIAM review, 26(2):195–239, 1984.   \n47 [51] H. Robbins and S. Monro. A stochastic approximation method. The annals of mathematical statistics, pages 400–407, 1951.   \n49 [52] C. Robert and G. Casella. Monte Carlo statistical methods. Springer Science & Business Media, 2013. [53] W. Siblini, P. Kuntz, and F. Meyer. A review on dimensionality reduction for multi-label classification. IEEE Transactions on Knowledge and Data Engineering, 2019.   \n53 [54] J. P. Siebert. Vehicle recognition using rule based methods. 1987. [55] C. Viroli and G. J. McLachlan. Deep gaussian mixture models. Statistics and Computing, 29(1):43–51, 2019.   \n56 [56] C. F. J. Wu. On the convergence properties of the EM algorithm. The Annals of statistics, pages 95–103, 1983.   \n58 [57] D. Wu and J. Ma. An effective EM algorithm for mixtures of Gaussian processes via the MCMC sampling and approximation. Neurocomputing, 331:366–374, 2019. ",
+        "bbox": [
+            155,
+            85,
+            830,
+            920
+        ],
+        "page_idx": 11
+    },
+    {
+        "type": "text",
+        "text": "60 [58] W. Xiang, A. Karfoul, C. Yang, H. Shu, and R. L. B. Jeannès. An exact line search scheme to   \n61 accelerate the EM algorithm: Application to Gaussian mixture models identification. Journal of   \n62 computational science, 41:101073, 2020.   \n63 [59] M. Zhang, Y. Fu, K. M. Bennett, and T. Wu. Computational efficient variational Bayesian Gaus  \n64 sian mixture models via coreset. In 2016 International Conference on Computer, Information   \n65 and Telecommunication Systems (CITS), pages 1–5. IEEE, 2016.   \n66 [60] H. Zhao, P. Poupart, and G. Gordon. A unified approach for learning the parameters of sum  \n67 product networks. In Proceedings of the 30th International Conference on Neural Information   \n68 Processing Systems, pages 433–441, 2016. ",
+        "bbox": [
+            150,
+            90,
+            826,
+            237
+        ],
+        "page_idx": 12
+    },
+    {
+        "type": "text",
+        "text": "1. For all authors... ",
+        "bbox": [
+            214,
+            116,
+            339,
+            131
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "(a) Do the main claims made in the abstract and introduction accurately reflect the paper’s contributions and scope? [Yes]   \n(b) Did you describe the limitations of your work? [Yes] Our main contribution is computational speedup. Cases where it was not achieved are highlighted in the experimental section.   \n(c) Did you discuss any potential negative societal impacts of your work? [No] We do not foresee any potential negative impact.   \n(d) Have you read the ethics review guidelines and ensured that your paper conforms to them? [Yes] ",
+        "bbox": [
+            238,
+            135,
+            825,
+            267
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "2. If you are including theoretical results... ",
+        "bbox": [
+            215,
+            272,
+            493,
+            286
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "(a) Did you state the full set of assumptions of all theoretical results? [N/A] (b) Did you include complete proofs of all theoretical results? [N/A] ",
+        "bbox": [
+            238,
+            290,
+            736,
+            321
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "3. If you ran experiments... ",
+        "bbox": [
+            212,
+            325,
+            393,
+            340
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "(a) Did you include the code, data, and instructions needed to reproduce the main experimental results (either in the supplemental material or as a URL)? [Yes] The code is available in a github repository. All dataset are public from the UCI database.   \n(b) Did you specify all the training details (e.g., data splits, hyperparameters, how they were chosen)? [Yes] See Section 6.   \n(c) Did you report error bars (e.g., with respect to the random seed after running experiments multiple times)? [No] We report only average of Monte Carlo repetitions, the error bars were too small to have any visual impact in the reported logarithmic scale.   \n(d) Did you include the total amount of compute and the type of resources used (e.g., type of GPUs, internal cluster, or cloud provider)? [Yes] ",
+        "bbox": [
+            238,
+            344,
+            825,
+            489
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "4. If you are using existing assets (e.g., code, data, models) or curating/releasing new assets... ",
+        "bbox": [
+            215,
+            494,
+            823,
+            510
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "(a) If your work uses existing assets, did you cite the creators? [Yes] We use 20 datasets from UCI, we cite the required papers for each dataset, mostly the UCI database and few additional publications.   \n(b) Did you mention the license of the assets? [No] The data are publically available, we comply with the requirement on citing appropriate publications.   \n(c) Did you include any new assets either in the supplemental material or as a URL? [No]   \n(d) Did you discuss whether and how consent was obtained from people whose data you’re using/curating? [N/A]   \n(e) Did you discuss whether the data you are using/curating contains personally identifiable information or offensive content? [N/A] ",
+        "bbox": [
+            238,
+            513,
+            825,
+            662
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "5. If you used crowdsourcing or conducted research with human subjects... ",
+        "bbox": [
+            215,
+            666,
+            707,
+            681
+        ],
+        "page_idx": 13
+    },
+    {
+        "type": "text",
+        "text": "(a) Did you include the full text of instructions given to participants and screenshots, if applicable? [N/A]   \n(b) Did you describe any potential participant risks, with links to Institutional Review Board (IRB) approvals, if applicable? [N/A]   \n(c) Did you include the estimated hourly wage paid to participants and the total amount spent on participant compensation? [N/A] ",
+        "bbox": [
+            238,
+            685,
+            825,
+            773
+        ],
+        "page_idx": 13
+    }
+]

parse/train/ZS394D3djsg/ZS394D3djsg_middle.json ADDED Viewed

The diff for this file is too large to render. See raw diff

parse/train/ZS394D3djsg/ZS394D3djsg_model.json ADDED Viewed

The diff for this file is too large to render. See raw diff

parse/train/hb1sDDSLbV/hb1sDDSLbV.md ADDED Viewed

The diff for this file is too large to render. See raw diff

parse/train/hb1sDDSLbV/hb1sDDSLbV_content_list.json ADDED Viewed

The diff for this file is too large to render. See raw diff

parse/train/hb1sDDSLbV/hb1sDDSLbV_middle.json ADDED Viewed

The diff for this file is too large to render. See raw diff

parse/train/hb1sDDSLbV/hb1sDDSLbV_model.json ADDED Viewed

The diff for this file is too large to render. See raw diff

parse/train/ry_WPG-A-/ry_WPG-A-.md ADDED Viewed

	@@ -0,0 +1,384 @@

+# ON THE INFORMATION BOTTLENECK THEORY OF DEEP LEARNING
+Andrew M. Saxe, Yamini Bansal, Joel Dapello, Madhu Advani
+Harvard University
+{asaxe,madvani}@fas.harvard.edu,{ybansal,dapello}@g.harvard.edu
+Artemy Kolchinsky, Brendan D. Tracey
+Santa Fe Institute {artemyk,tracey.brendan}@gmail.com
+David D. Cox
+Harvard University
+MIT-IBM Watson AI Lab
+davidcox@fas.harvard.edu
+david.d.cox@ibm.com
+# ABSTRACT
+The practical successes of deep neural networks have not been matched by theoretical progress that satisfyingly explains their behavior. In this work, we study the information bottleneck (IB) theory of deep learning, which makes three specific claims: first, that deep networks undergo two distinct phases consisting of an initial fitting phase and a subsequent compression phase; second, that the compression phase is causally related to the excellent generalization performance of deep networks; and third, that the compression phase occurs due to the diffusion-like behavior of stochastic gradient descent. Here we show that none of these claims hold true in the general case. Through a combination of analytical results and simulation, we demonstrate that the information plane trajectory is predominantly a function of the neural nonlinearity employed: double-sided saturating nonlinearities like tanh yield a compression phase as neural activations enter the saturation regime, but linear activation functions and single-sided saturating nonlinearities like the widely used ReLU in fact do not. Moreover, we find that there is no evident causal connection between compression and generalization: networks that do not compress are still capable of generalization, and vice versa. Next, we show that the compression phase, when it exists, does not arise from stochasticity in training by demonstrating that we can replicate the IB findings using full batch gradient descent rather than stochastic gradient descent. Finally, we show that when an input domain consists of a subset of task-relevant and task-irrelevant information, hidden representations do compress the task-irrelevant information, although the overall information about the input may monotonically increase with training time, and that this compression happens concurrently with the fitting process rather than during a subsequent compression period.
+# 1 INTRODUCTION
+Deep neural networks (Schmidhuber, 2015; LeCun et al., 2015) are the tool of choice for real-world tasks ranging from visual object recognition (Krizhevsky et al., 2012), to unsupervised learning (Goodfellow et al., 2014; Lotter et al., 2016) and reinforcement learning (Silver et al., 2016). These practical successes have spawned many attempts to explain the performance of deep learning systems (Kadmon & Sompolinsky, 2016), mostly in terms of the properties and dynamics of the optimization problem in the space of weights (Saxe et al., 2014; Choromanska et al., 2015; Advani & Saxe, 2017), or the classes of functions that can be efficiently represented by deep networks (Montufar et al., 2014; Poggio et al., 2017). This paper analyzes a recent inventive proposal to study the dynamics of learning through the lens of information theory (Tishby & Zaslavsky, 2015; Shwartz-Ziv & Tishby, 2017). In this view, deep learning is a question of representation learning: each layer of a deep neural network can be seen as a set of summary statistics which contain some but not all of the information present in the input, while retaining as much information about the target output as possible. The amount of information in a hidden layer regarding the input and output can then be measured over the course of learning, yielding a picture of the optimization process in the information plane. Crucially, this method holds the promise to serve as a general analysis that can be used to compare different architectures, using the common currency of mutual information. Moreover, the elegant information bottleneck (IB) theory provides a fundamental bound on the amount of input compression and target output information that any representation can achieve (Tishby et al., 1999). The IB bound thus serves as a method-agnostic ideal to which different architectures and algorithms may be compared.
+A preliminary empirical exploration of these ideas in deep neural networks has yielded striking findings (Shwartz-Ziv & Tishby, 2017). Most saliently, trajectories in the information plane appear to consist of two distinct phases: an initial “fitting” phase where mutual information between the hidden layers and both the input and output increases, and a subsequent “compression” phase where mutual information between the hidden layers and the input decreases. It has been hypothesized that this compression phase is responsible for the excellent generalization performance of deep networks, and further, that this compression phase occurs due to the random diffusion-like behavior of stochastic gradient descent.
+Here we study these phenomena using a combination of analytical methods and simulation. In Section 2, we show that the compression observed by Shwartz-Ziv & Tishby (2017) arises primarily due to the double-saturating tanh activation function used. Using simple models, we elucidate the effect of neural nonlinearity on the compression phase. Importantly, we demonstrate that the ReLU activation function, often the nonlinearity of choice in practice, does not exhibit a compression phase. We discuss how this compression via nonlinearity is related to the assumption of binning or noise in the hidden layer representation. To better understand the dynamics of learning in the information plane, in Section 3 we study deep linear networks in a tractable setting where the mutual information can be calculated exactly. We find that deep linear networks do not compress over the course of training for the setting we examine. Further, we show a dissociation between generalization and compression. In Section 4, we investigate whether stochasticity in the training process causes compression in the information plane. We train networks with full batch gradient descent, and compare the results to those obtained with stochastic gradient descent. We find comparable compression in both cases, indicating that the stochasticity of SGD is not a primary factor in the observed compression phase. Moreover, we show that the two phases of SGD occur even in networks that do not compress, demonstrating that the phases are not causally related to compression. These results may seem difficult to reconcile with the intuition that compression can be necessary to attain good performance: if some input channels primarily convey noise, good generalization requires excluding them. Therefore, in Section 5 we study a situation with explicitly task-relevant and task-irrelevant input dimensions. We show that the hidden-layer mutual information with the task-irrelevant subspace does indeed drop during training, though the overall information with the input increases. However, instead of a secondary compression phase, this task-irrelevant information is compressed at the same time that the taskrelevant information is boosted. Our results highlight the importance of noise assumptions in applying information theoretic analyses to deep learning systems, and put in doubt the generality of the IB theory of deep learning as an explanation of generalization performance in deep architectures.
+# 2 COMPRESSION AND NEURAL NONLINEARITIES
+The starting point for our analysis is the observation that changing the activation function can markedly change the trajectory of a network in the information plane. In Figure 1A, we show our replication of the result reported by Shwartz-Ziv & Tishby (2017) for networks with the tanh nonlinearity.1 This replication was performed with the code supplied by the authors of Shwartz-Ziv & Tishby (2017), and closely follows the experimental setup described therein. Briefly, a neural network with 7 fully connected hidden layers of width 12-10-7-5-4-3-2 is trained with stochastic gradient descent to produce a binary classification from a 12-dimensional input. In our replication we used 256 randomly selected samples per batch. The mutual information of the network layers with respect to the input and output variables is calculated by binning the neuron’s tanh output activations into 30 equal intervals between -1 and 1. Discretized values for each neuron in each layer are then used to directly calculate the joint distributions, over the 4096 equally likely input patterns and true output labels. In line with prior work (Shwartz-Ziv & Tishby, 2017), the dynamics in Fig. 1 show a transition between an initial fitting phase, during which information about the input increases, and a subsequent compression phase, during which information about the input decreases.
+![](images/ae057461c9856b2cafbb8dc433b0cc3cacf699acd8de55f5ace722373b08a275.jpg)
+Figure 1: Information plane dynamics and neural nonlinearities. (A) Replication of Shwartz-Ziv & Tishby (2017) for a network with tanh nonlinearities (except for the final classification layer which contains two sigmoidal neurons). The $\mathbf { X }$ -axis plots information between each layer and the input, while the y-axis plots information between each layer and the output. The color scale indicates training time in epochs. Each of the six layers produces a curve in the information plane with the input layer at far right, output layer at the far left. Different layers at the same epoch are connected by fine lines. (B) Information plane dynamics with ReLU nonlinearities (except for the final layer of 2 sigmoidal neurons). Here no compression phase is visible in the ReLU layers. For learning curves of both networks, see Appendix A. (C) Information plane dynamics for a tanh network of size $7 8 4 - 1 0 2 4 - 2 0 - 2 0 - 2 0 - 1 0$ trained on MNIST, estimated using the non-parametric kernel density mutual information estimator of Kolchinsky & Tracey (2017); Kolchinsky et al. (2017), no compression is observed except in the final classification layer with sigmoidal neurons. See Appendix B for the KDE MI method applied to the original Tishby dataset; additional results using a second popular nonparametric $\mathbf { k }$ -NN-based method (Kraskov et al., 2004); and results for other neural nonlinearities.
+We then modified the code to train deep networks using rectified linear activation functions $( f ( x ) =$ $\operatorname* { m a x } ( 0 , x ) )$ . While the activities of tanh networks are bounded in the range $[ - 1 , 1 ]$ , ReLU networks have potentially unbounded positive activities. To calculate mutual information, we first trained the ReLU networks, next identified their largest activity value over the course of training, and finally chose 100 evenly spaced bins between the minimum and maximum activity values to discretize the hidden layer activity. The resulting information plane dynamics are shown in Fig. 1B. The mutual information with the input monotonically increases in all ReLU layers, with no apparent compression phase. To see whether our results were an artifact of the small network size, toy dataset, or simple binning-based mutual information estimator we employed, we also trained larger networks on the MNIST dataset and computed mutual information using a state-of-the-art nonparametric kernel density estimator which assumes hidden activity is distributed as a mixture of Gaussians (see Appendix B for details). Fig. C-D show that, again, tanh networks compressed but ReLU networks did not. Appendix B shows that similar results also obtain with the popular nonparametric $\mathbf { k }$ -nearest-neighbor estimator of Kraskov et al. (2004), and for other neural nonlinearities. Thus, the choice of nonlinearity substantively affects the dynamics in the information plane.
+To understand the impact of neural nonlinearity on the mutual information dynamics, we develop a minimal model that exhibits this phenomenon. In particular, consider the simple three neuron network shown in Fig. 2A. We assume a scalar Gaussian input distribution $X \sim \mathcal { N } ( 0 , 1 )$ , which is fed through the scalar first layer weight $w _ { 1 }$ , and passed through a neural nonlinearity $f ( \cdot )$ , yielding the hidden unit activity $h = f ( w _ { 1 } X )$ . To calculate the mutual information with the input, this hidden unit activity is then binned yielding the new discrete variable $T = \dot { \mathbf { b i n } } ( h )$ (for instance, into 30 evenly spaced bins from $^ { - 1 }$ to 1 for the tanh nonlinearity). This binning process is depicted in Fig. 2B. In this simple setting, the mutual information $I ( T ; X )$ between the binned hidden layer activity $T$ and the input $X$ can be calculated exactly. In particular,
+$$
+\begin{array} { l c l } { { { \cal I } ( T ; X ) } } & { { = } } & { { { \cal H } ( T ) - { \cal H } ( T | X ) } } \\ { { } } & { { = } } & { { { \cal H } ( T ) } } \\ { { } } & { { = } } & { { - \displaystyle \sum _ { i = 1 } ^ { N } p _ { i } \log p _ { i } } } \end{array}
+$$
+where $H ( \cdot )$ denotes entropy, and we have used the fact that $H ( T | X ) = 0$ since $T$ is a deterministic function of $X$ . Here the probabilities $p _ { i } = P ( h \ge b _ { i }$ and $h < b _ { i + 1 }$ ) are simply the probability that an input $X$ produces a hidden unit activity that lands in bin $i$ , defined by lower and upper bin limits $b _ { i }$ and $b _ { i + 1 }$ respectively. This probability can be calculated exactly for monotonic nonlinearities $f ( \cdot )$ using the cumulative density of $X$ ,
+$$
+p _ { i } = P ( X \geq f ^ { - 1 } ( b _ { i } ) / w _ { 1 } \mathrm { a n d } X < f ^ { - 1 } ( b _ { i + 1 } ) / w _ { 1 } ) ,
+$$
+where $f ^ { - 1 } ( \cdot )$ is the inverse function of $f ( \cdot )$ .
+As shown in Fig. 2C-D, as a function of the weight $w _ { 1 }$ , mutual information with the input first increases and then decreases for the tanh nonlinearity, but always increases for the ReLU nonlinearity. Intuitively, for small weights $w _ { 1 } \approx 0$ , neural activities lie near zero on the approximately linear part of the tanh function. Therefore $f ( w _ { 1 } X ) \approx w _ { 1 } X$ , yielding a rescaled Gaussian with information that grows with the size of the weights. However for very large weights $w _ { 1 } \to \infty$ , the tanh hidden unit nearly always saturates, yielding a discrete variable that concentrates in just two bins. This is more or less a coin flip, containing mutual information with the input of approximately 1 bit. Hence the distribution of $T$ collapses to a much lower entropy distribution, yielding compression for large weight values. With the ReLU nonlinearity, half of the inputs are negative and land in the bin containing a hidden activity of zero. The other half are Gaussian distributed, and thus have entropy that increases with the size of the weight.
+Hence double-saturating nonlinearities can lead to compression of information about the input, as hidden units enter their saturation regime, due to the binning procedure used to calculate mutual information. The crux of the issue is that the actual $I ( h ; X )$ is infinite, unless the network itself adds noise to the hidden layers. In particular, without added noise, the transformation from $X$ to the continuous hidden activity $h$ is deterministic and the mutual information $I ( h ; X )$ would generally be infinite (see Appendix C for extended discussion). Networks that include noise in their processing (e.g., Kolchinsky et al. (2017)) can have finite $I ( T ; X )$ . Otherwise, to obtain a finite MI, one must compute mutual information as though there were binning or added noise in the activations. But this binning/noise is not actually a part of the operation of the network, and is therefore somewhat arbitrary (different binning schemes can result in different mutual information with the input, as shown in Fig. 14 of Appendix C).
+![](images/b7f23d4571d8a39c6d37d9f1a3d373f749c5196988ea773d81a640c8c050f81d.jpg)
+Figure 2: Nonlinear compression in a minimal model. (A) A three neuron nonlinear network which receives Gaussian inputs $x$ , multiplies by weight $w _ { 1 }$ , and maps through neural nonlinearity $f ( \cdot )$ to produce hidden unit activity $h$ . (B) The continuous activity $h$ is binned into a discrete variable $T$ for the purpose of calculating mutual information. Blue: continuous tanh nonlinear activation function. Grey: Bin borders for 30 bins evenly spaced between $^ { - 1 }$ and 1. Because of the saturation in the sigmoid, a wide range of large magnitude net input values map to the same bin. (C) Mutual information with the input as a function of weight size $w _ { 1 }$ for a tanh nonlinearity. Information increases for small $w _ { 1 }$ and then decreases for large $w _ { 1 }$ as all inputs land in one of the two bins corresponding to the saturation regions. (D) Mutual information with the input for the ReLU nonlinearity increases without bound. Half of all inputs land in the bin corresponding to zero activity, while the other half have information that scales with the size of the weights.
+We note that the binning procedure can be viewed as implicitly adding noise to the hidden layer activity: a range of $X$ values map to a single bin, such that the mapping between $X$ and $T$ is no longer perfectly invertible (Laughlin, 1981). The binning procedure is therefore crucial to obtaining a finite MI value, and corresponds approximately to a model where noise enters the system after the calculation of $h$ , that is, $T = h + \epsilon$ , where $\epsilon$ is noise of fixed variance independent from $h$ and $X$ . This approach is common in information theoretic analyses of deterministic systems, and can serve as a measure of the complexity of a system’s representation (see Sec 2.4 of Shwartz-Ziv & Tishby (2017)). However, neither binning nor noise is present in the networks that Shwartz-Ziv & Tishby (2017) considered, nor the ones in Fig. 2, either during training or testing. It therefore remains unclear whether robustness of a representation to this sort of noise in fact influences generalization performance in deep learning systems.
+Furthermore, the addition of noise means that different architectures may no longer be compared in a common currency of mutual information: the binning/noise structure is arbitrary, and architectures that implement an identical input-output map can nevertheless have different robustness to noise added in their internal representation. For instance, Appendix C describes a family of linear networks that compute exactly the same input-output map and therefore generalize identically, but yield different mutual information with respect to the input. Finally, we note that approaches which view the weights obtained from the training process as the random variables of interest may sidestep this issue (Achille & Soatto, 2017).
+![](images/b90f7481a85a83f3aa2ebfdd7fb9a3e639595486fdd68886cf2cb42a1f704c6f.jpg)
+Figure 3: Generalization and information plane dynamics in deep linear networks. (A) A linear teacher network generates a dataset by passing Gaussian inputs $X$ through its weights and adding noise. (B) A deep linear student network is trained on the dataset (here the network has 1 hidden layer to allow comparison with Fig. 4A, see Supplementary Figure 18 for a deeper network). (C) Training and testing error over time. (D) Information plane dynamics. No compression is observed.
+Hence when a tanh network is initialized with small weights and over the course of training comes to saturate its nonlinear units (as it must to compute most functions of practical interest, see discussion in Appendix D), it will enter a compression period where mutual information decreases. Figures 16-17 of Appendix E show histograms of neural activity over the course of training, demonstrating that activities in the tanh network enter the saturation regime during training. This nonlinearity-based compression furnishes another explanation for the observation that training slows down as tanh networks enter their compression phase (Shwartz-Ziv & Tishby, 2017): some fraction of inputs have saturated the nonlinearities, reducing backpropagated error gradients.
+# 3 INFORMATION PLANE DYNAMICS IN DEEP LINEAR NETWORKS
+The preceding section investigates the role of nonlinearity in the observed compression behavior, tracing the source to double-saturating nonlinearities and the binning methodology used to calculate mutual information. However, other mechanisms could lead to compression as well. Even without nonlinearity, neurons could converge to highly correlated activations, or project out irrelevant directions of the input. These phenomena are not possible to observe in our simple three neuron minimal model, as they require multiple inputs and hidden layer activities. To search for these mechanisms, we turn to a tractable model system: deep linear neural networks (Baldi & Hornik (1989); Fukumizu (1998); Saxe et al. (2014)). In particular, we exploit recent results on the generalization dynamics in simple linear networks trained in a student-teacher setup (Seung et al., 1992; Advani & Saxe, 2017). In a student-teacher setting, one “student” neural network learns to approximate the output of another “teacher” neural network. This setting is a way of generating a dataset with interesting structure that nevertheless allows exact calculation of the generalization performance of the network, exact calculation of the mutual information of the representation (without any binning procedure), and, though we do not do so here, direct comparison to the IB bound which is already known for linear Gaussian problems (Chechik et al., 2005).
+We consider a scenario where a linear teacher neural network generates input and output examples which are then fed to a deep linear student network to learn (Fig. 3A). Following the formulation of (Advani & Saxe, 2017), we assume multivariate Gaussian inputs $\begin{array} { r } { X \sim \mathcal { N } ( 0 , \frac { 1 } { N _ { i } } \overline { { I } } _ { N _ { i } } ) } \end{array}$ and a scalar output $Y$ . The output is generated by the teacher network according to $Y = \dot { W } _ { 0 } X + \epsilon _ { o }$ , where $\epsilon _ { o } \stackrel { - } { \sim } \mathcal { N } ( 0 , \sigma _ { o } ^ { 2 } )$ represents aspects of the target function which cannot be represented by a neural network (that is, the approximation error or bias in statistical learning theory), and the teacher weights $W _ { o }$ are drawn independently from ${ \mathcal N } ( 0 , \sigma _ { w } ^ { 2 } )$ . Here, the weights of the teacher define the rule to be learned. The signal to noise ratio $\mathrm { S N R } = \sigma _ { w } ^ { 2 } / \sigma _ { o } ^ { 2 }$ determines the strength of the rule linking inputs to outputs relative to the inevitable approximation error. We emphasize that the “noise” added to the teacher’s output is fundamentally different from the noise added for the purpose of calculating mutual information: $\epsilon _ { o }$ models the approximation error for the task–even the best possible neural network may still make errors because the target function is not representable exactly as a neural network–and is part of the construction of the dataset, not part of the analysis of the student network.
+To train the student network, a dataset of $P$ examples is generated using the teacher. The student network is then trained to minimize the mean squared error between its output and the target output using standard (batch or stochastic) gradient descent on this dataset. Here the student is a deep linear neural network consisting of potentially many layers, but where the the activation function of each neuron is simply $f ( u ) = u$ . That is, a depth $D$ deep linear network computes the output $\hat { Y } = W _ { D + 1 } W _ { D } \cdot \cdot \cdot W _ { 2 } W _ { 1 } X$ . While linear activation functions stop the network from computing complex nonlinear functions of the input, deep linear networks nevertheless show complicated nonlinear learning trajectories (Saxe et al., 2014), the optimization problem remains nonconvex (Baldi & Hornik, 1989), and the generalization dynamics can exhibit substantial overtraining (Fukumizu, 1998; Advani & Saxe, 2017).
+Importantly, because of the simplified setting considered here, the true generalization error is easily shown to be
+$$
+E _ { g } ( t ) = | | W _ { o } - W _ { t o t } ( t ) | | _ { F } ^ { 2 } + \sigma _ { o } ^ { 2 }
+$$
+where $W _ { t o t } ( t )$ is the overall linear map implemented by the network at training epoch $t$ (that is, $W _ { t o t } = W _ { D + 1 } W _ { D } \cdot \cdot \cdot W _ { 2 } W _ { 1 } )$ .
+Furthermore, the mutual information with the input and output may be calculated exactly, because the distribution of the activity of any hidden layer is Gaussian. Let $T$ be the activity of a specific hidden layer, and let $\bar { W }$ be the linear map from the input to this activity (that is, for layer $l$ , $\bar { W } =$ $W _ { l } \cdots W _ { 2 } W _ { 1 } )$ . Since $T = { \bar { W } } X$ , the mutual information of $X$ and $T$ calculated using differential entropy is infinite. For the purpose of calculating the mutual information, therefore, we assume that Gaussian noise is added to the hidden layer activity, $T = \bar { W } X + \epsilon _ { M I }$ , with mean 0 and variance $\sigma _ { M I } ^ { 2 } = 1 . 0$ . This allows the analysis to apply to networks of any size, including overcomplete layers, but as before we emphasize that we do not add this noise either during training or testing. With these assumptions, $T$ and $X$ are jointly Gaussian and we have
+$$
+I ( T ; X ) = \log \lvert \bar { W } \bar { W } ^ { T } + \sigma _ { M I } ^ { 2 } I _ { N _ { h } } \rvert - \log \lvert \sigma _ { M I } ^ { 2 } I _ { N _ { h } } \rvert
+$$
+where $\left| \cdot \right|$ denotes the determinant of a matrix. Finally the mutual information with the output $Y$ , also jointly Gaussian, can be calculated similarly (see Eqns. (22)-(25) of Appendix G).
+Fig. 3 shows example training and test dynamics over the course of learning in panel C, and the information plane dynamics in panel D. Here the network has an input layer of 100 units, 1 hidden layer of 100 units each and one output unit. The network was trained with batch gradient descent on a dataset of 100 examples drawn from the teacher with signal to noise ratio of 1.0. The linear network behaves qualitatively like the ReLU network, and does not exhibit compression. Nevertheless, it learns a map that generalizes well on this task and shows minimal overtraining. Hence, in the setting we study here, generalization performance can be acceptable without any compression phase.
+The results in (Advani & Saxe (2017)) show that, for the case of linear networks, overtraining is worst when the number of inputs matches the number of training samples, and is reduced by making the number of samples smaller or larger. Fig. 4 shows learning dynamics with the number of samples matched to the size of the network. Here overfitting is substantial, and again no compression is seen in the information plane. Comparing to the result in Fig. 3D, both networks exhibit similar information dynamics with respect to the input (no compression), but yield different generalization performance.
+Hence, in this linear analysis of a generic setting, there do not appear to be additional mechanisms that cause compression over the course of learning; and generalization behavior can be widely different for networks with the same dynamics of information compression regarding the input. We note that, in the setting considered here, all input dimensions have the same variance, and the weights of the teacher are drawn independently. Because of this, there are no special directions in the input, and each subspace of the input contains as much information as any other. It is possible that, in real world tasks, higher variance inputs are also the most likely to be relevant to the task (here, have large weights in the teacher). We have not investigated this possibility here.
+![](images/22b7932d2a586116824d010622a5076fd367f6f9658cd0958fd6f2a3a8c1187a.jpg)
+Figure 4: Overtraining and information plane dynamics. (A) Average training and test mean square error for a deep linear network trained with SGD. Overtraining is substantial. Other parameters: $N _ { i } =$ 100, $\mathrm { P } = 1 0 0$ , Number of hidden units $= 1 0 0$ , Batch size $= 5$ (B) Information plane dynamics. No compression is observed, and information about the labels is lost during overtraining. (C) Average train and test accuracy $\%$ correct) for nonlinear tanh networks exhibiting modest overfitting $N = 8$ ). (D) Information plane dynamics. Overfitting occurs despite continued compression.
+![](images/6fb26702a9042ab8cf86aa3fd585a3fceb5522ab227a6c80568e9d5d17d27d7b.jpg)
+Figure 5: Stochastic training and the information plane. (A) tanh network trained with SGD. (B) tanh network trained with BGD. (C) ReLU network trained with SGD. (D) ReLU network trained with BGD. Both random and non-random training procedures show similar information plane dynamics.
+To see whether similar behavior arises in nonlinear networks, we trained tanh networks in the same setting as Section 2, but with $30 \%$ of the data, which we found to lead to modest overtraining. Fig. 4C-D shows the resulting train, test, and information plane dynamics. Here the tanh networks show substantial compression, despite exhibiting overtraining. This establishes a dissociation between behavior in the information plane and generalization dynamics: networks that compress may (Fig. 1A) or may not (Fig. 4C-D) generalize well, and networks that do not compress may (Figs.1B, 3A-B) or may not (Fig. 4A-B) generalize well.
+# 4 COMPRESSION IN BATCH GRADIENT DESCENT AND SGD
+Next, we test a core theoretical claim of the information bottleneck theory of deep learning, namely that randomness in stochastic gradient descent is responsible for the compression phase. In particular, because the choice of input samples in SGD is random, the weights evolve in a stochastic way during training.
+Shwartz-Ziv & Tishby (2017) distinguish two phases of SGD optimization: in the first “drift” phase, the mean of the gradients over training samples is large relative to the standard deviation of the gradients; in the second “diffusion” phase, the mean becomes smaller than the standard deviation of the gradients. The authors propose that compression should commence following the transition from a high to a low gradient signal-to-noise ratio (SNR), i.e., the onset of the diffusion phase. The proposed mechanism behind this diffusion-driven compression is as follows. The authors state that during the diffusion phase, the stochastic evolution of the weights can be described as a Fokker-Planck equation under the constraint of small training error. Then, the stationary distribution over weights for this process will have maximum entropy, again subject to the training error constraint. Finally, the authors claim that weights drawn from this stationary distribution will maximize the entropy of inputs given hidden layer activity, $H ( X | T )$ , subject to a training error constraint, and that this training error constraint is equivalent to a constraint on the mutual information $I ( T ; Y )$ for small training error. Since the entropy of the input, $H ( X )$ , is fixed, the result of the diffusion dynamics will be to minimize $I ( X ; T ) \mathrel { \mathop : } = H ( X ) - \bar { H } ( X | \dot { T } )$ for a given value of $I ( T ; Y )$ reached at the end of the drift phase.
+However, this explanation does not hold up to either theoretical or empirical investigation. Let us assume that the diffusion phase does drive the distribution of weights to a maximum entropy distribution subject to a training error constraint. Note that this distribution reflects stochasticity of weights across different training runs. There is no general reason that a given set of weights sampled from this distribution (i.e., the weight parameters found in one particular training run) will maximize $H ( X | T )$ , the entropy of inputs given hidden layer activity. In particular, $H ( X | T )$ reflects (conditional) uncertainty about inputs drawn from the data-generating distribution, rather than uncertainty about any kind of distribution across different training runs.
+We also show empirically that the stochasticity of the SGD is not necessary for compression. To do so, we consider two distinct training procedures: offline stochastic gradient descent (SGD), which learns from a fixed-size dataset, and updates weights by repeatedly sampling a single example from the dataset and calculating the gradient of the error with respect to that single sample (the typical procedure used in practice); and batch gradient descent (BGD), which learns from a fixed-size dataset, and updates weights using the gradient of the total error across all examples. Batch gradient descent uses the full training dataset and, crucially, therefore has no randomness or diffusion-like behavior in its updates.
+We trained tanh and ReLU networks with SGD and BGD and compare their information plane dynamics in Fig. 5 (see Appendix H for a linear network). We find largely consistent information dynamics in both instances, with robust compression in tanh networks for both methods. Thus randomness in the training process does not appear to contribute substantially to compression of information about the input. This finding is consistent with the view presented in Section 2 that compression arises predominantly from the double saturating nonlinearity.
+Finally, we look at the gradient signal-to-noise ratio (SNR) to analyze the relationship between compression and the transition from high to low gradient SNR. Fig. 20 of Appendix I shows the gradient SNR over training, which in all cases shows a phase transition during learning. Hence the gradient SNR transition is a general phenomenon, but is not causally related to compression. Appendix I offers an extended discussion and shows gradient SNR transitions without compression on the MNIST dataset and for linear networks.
+# 5 SIMULTANEOUS FITTING AND COMPRESSION
+Our finding that generalization can occur without compression may seem difficult to reconcile with the intuition that certain tasks involve suppressing irrelevant directions of the input. In the extreme, if certain inputs contribute nothing but noise, then good generalization requires ignoring them. To study this, we consider a variant on the linear student-teacher setup of Section 3: we partition the input $X$ into a set of task-relevant inputs $X _ { r e l }$ and a set of task-irrelevant inputs $X _ { i r r e l }$ , and alter the teacher network so that the teacher’s weights to the task-irrelevant inputs are all zero. Hence the inputs $X _ { i r r e l }$ contribute only noise, while the $X _ { r e l }$ contain signal. We then calculate the information plane dynamics for the whole layer, and for the task-relevant and task-irrelevant inputs separately. Fig. 6 shows information plane dynamics for a deep linear neural network trained using SGD (5 samples/batch) on a task with 30 task-relevant inputs and 70 task-irrelevant inputs. While the overall dynamics show no compression phase, the information specifically about the task-irrelevant subspace does compress over the course of training. This compression process occurs at the same time as the fitting to the task-relevant information. Thus, when a task requires ignoring some inputs, the information with these inputs specifically will indeed be reduced; but overall mutual information with the input in general may still increase.
+![](images/3f982bb2a95db7292dd21c09073b2ac2ac7ccf908216d9ad8ed4dcda065e419e.jpg)
+Figure 6: Simultaneous fitting and compression. (A) For a task with a large task-irrelevant subspace in the input, a linear network shows no overall compression of information about the input. (B) The information with the task-relevant subspace increases robustly over training. (C) However, the information specifically about the task-irrelevant subspace does compress after initially growing as the network is trained.
+# 6 DISCUSSION
+Our results suggest that compression dynamics in the information plane are not a general feature of deep networks, but are critically influenced by the nonlinearities employed by the network. Doublesaturating nonlinearities lead to compression, if mutual information is estimated by binning activations or by adding homoscedastic noise, while single-sided saturating nonlinearities like ReLUs do not compress in general. Consistent with this view, we find that stochasticity in the training process does not contribute to compression in the cases we investigate. Furthermore, we have found instances where generalization performance does not clearly track information plane behavior, questioning the causal link between compression and generalization. Hence information compression may parallel the situation with sharp minima: although empirical evidence has shown a correlation with generalization error in certain settings and architectures, further theoretical analysis has shown that sharp minima can in fact generalize well (Dinh et al., 2017). We emphasize that compression still may occur within a subset of the input dimensions if the task demands it. This compression, however, is interleaved rather than in a secondary phase and may not be visible by information metrics that track the overall information between a hidden layer and the input. Finally, we note that our results address the specific claims of one scheme to link the information bottleneck principle with current practice in deep networks. The information bottleneck principle itself is more general and may yet offer important insights into deep networks (Achille & Soatto, 2017). Moreover, the information bottleneck principle could yield fundamentally new training algorithms for networks that are inherently stochastic and where compression is explicitly encouraged with appropriate regularization terms (Chalk et al., 2016; Alemi et al., 2017; Kolchinsky et al., 2017).
+# ACKNOWLEDGMENTS
+We thank Ariel Herbert-Voss for useful discussions. This work was supported by grant numbers IIS 1409097 and CHE 1648973 from the US National Science Foundation, and by IARPA contract #D16PC00002. Andrew Saxe and Madhu Advani thank the Swartz Program in Theoretical Theoretical Neuroscience at Harvard University. Artemy Kolchinsky and Brendan Tracey would like to thank the Santa Fe Institute for helping to support this research. Artemy Kolchinsky was supported by Grant No. FQXi-RFP-1622 from the FQXi foundation and Grant No. CHE-1648973 from the US National Science Foundation. Brendan Tracey was supported by AFOSR MURI on Multi-Information Sources of Multi-Physics Systems under Award Number FA9550-15-1-0038.
+# REFERENCES
+A. Achille and S. Soatto. On the Emergence of Invariance and Disentangling in Deep Representations. arXiv preprint arXiv:1706.01350, 2017.
+M.S. Advani and A.M. Saxe. High-dimensional dynamics of generalization error in neural networks. arXiv preprint arXiv:1710.03667, 2017.
+A.A. Alemi, I. Fischer, J.V. Dillon, and K. Murphy. Deep variational information bottleneck. In International Conference on Learning Representations, 2017.
+P. Baldi and K. Hornik. Neural networks and principal component analysis: Learning from examples without local minima. Neural Networks, 2:53–58, 1989.
+P.L. Bartlett and S. Mendelson. Rademacher and Gaussian Complexities: Risk Bounds and Structural Results. Journal of Machine Learning Research, 3:463–482, 2002.
+M. Chalk, O. Marre, and G. Tkacik. Relevant sparse codes with variational information bottleneck. In Advances in Neural Information Processing Systems, pp. 1957–1965, 2016.
+G. Chechik, A. Globerson, N. Tishby, and Y. Weiss. Information bottleneck for gaussian variables. Journal of Machine Learning Research, pp. 165–188, 2005.
+J. Chee and P. Toulis. Convergence diagnostics for stochastic gradient descent with constant step size. arXiv preprint arXiv:1710.06382, 2017.
+A. Choromanska, M. Henaff, M. Mathieu, G. Arous, B., and Y. LeCun. The Loss Surfaces of Multilayer Networks. In Proceedings of the 18th International Conference on Artificial Intelligence, volume 38, 2015.
+L. Dinh, R. Pascanu, S. Bengio, and Y. Bengio. Sharp Minima Can Generalize For Deep Nets. In International Conference on Machine Learning, 2017.
+K. Fukumizu. Effect of Batch Learning In Multilayer Neural Networks. In Proceedings of the 5th International Conference on Neural Information Processing, pp. 67–70, 1998.
+I. Goodfellow, J. Pouget-Abadie, M. Mirza, B. Xu, D. Warde-Farley, S. Ozair, A. Courville, and Y. Bengio. Generative Adversarial Nets. Advances in Neural Information Processing Systems, pp. 2672–2680, 2014.
+J. Kadmon and H. Sompolinsky. Optimal Architectures in a Solvable Model of Deep Networks. In Advances in Neural Information Processing Systems, 2016.
+A. Kolchinsky and B.D. Tracey. Estimating mixture entropy with pairwise distances. Entropy, 19, 2017.
+A. Kolchinsky, B.D. Tracey, and D.H. Wolpert. Nonlinear information bottleneck. arXiv preprint arXiv:1705.02436, 2017.
+A. Kraskov, H. Stögbauer, and P. Grassberger. Estimating mutual information. Physical Review E, 69:066138, 2004.
+A. Krizhevsky, I. Sutskever, and G.E. Hinton. ImageNet classification with deep convolutional neural networks. In Advances in Neural Information Processing Systems, pp. 1097–1105. 2012.
+S. Laughlin. A simple coding procedure enhances a neuron’s information capacity. Zeitschrift für Naturforschung c, 36:910–912, 1981.
+Y. LeCun, Y. Bengio, and G.E. Hinton. Deep learning. Nature, 521:436–444, 2015.
+W. Lotter, G. Kreiman, and D.D. Cox. Deep predictive coding networks for video prediction and unsupervised learning. arXiv preprint arXiv:1605.08104, 2016.
+G. Montufar, R. Pascanu, K. Cho, and Y. Bengio. On the Number of Linear Regions of Deep Neural Networks. In Advances in Neural Information Processing Systems, 2014.
+N. Murata. A statistical study of on-line learning. In On-line Learning in Neural Networks, pp. 63–92. Cambridge University Press, 1998.
+B. Neyshabur, R. Tomioka, and N. Srebro. Norm-Based Capacity Control in Neural Networks. In Proceedings of The 28th Conference on Learning Theory, volume 40, pp. 1–26, 2015.
+H. Poggio, T.and Mhaskar, L. Rosasco, B. Miranda, and Q. Liao. Why and when can deep-but not shallow-networks avoid the curse of dimensionality: A review. International Journal of Automation and Computing, pp. 1–17, 2017.
+A.M. Saxe, J.L. McClelland, and S. Ganguli. Exact solutions to the nonlinear dynamics of learning in deep linear neural networks. In the International Conference on Learning Representations, 2014.
+J. Schmidhuber. Deep Learning in Neural Networks: An Overview. Neural Networks, 61:85–117, 2015.
+H.S. Seung, H. Sompolinsky, and N. Tishby. Statistical mechanics of learning from examples. Physical Review A, 45:6056–6091, 1992.
+R. Shwartz-Ziv and N. Tishby. Opening the black box of deep neural networks via information. arXiv preprint arXiv:1703.00810, 2017.
+D. Silver, A. Huang, C.J. Maddison, A. Guez, L. Sifre, G. van den Driessche, J. Schrittwieser, I. Antonoglou, V. Panneershelvam, M. Lanctot, S. Dieleman, D. Grewe, J. Nham, N. Kalchbrenner, I. Sutskever, T. Lillicrap, M. Leach, K. Kavukcuoglu, T. Graepel, and D. Hassabis. Mastering the game of Go with deep neural networks and tree search. Nature, 529:484–489, 2016.
+N. Tishby and N. Zaslavsky. Deep learning and the information bottleneck principle. In IEEE Information Theory Workshop, 2015.
+N. Tishby, F.C. Pereira, and W. Bialek. The information bottleneck method. Proceedings of the 37-th Annual Allerton Conference on Communication, Control and Computing, pp. 368–377, 1999.
+# A LEARNING CURVES FOR tanh AND RELU NETWORKS
+Supplementary Figure 7 shows the learning curves for tanh and ReLU networks depicted in Fig. 1.
+![](images/ce0355c43f7ded6cd043397ccb1f76e1f4c9361219e3e513122d676b64034395.jpg)
+Figure 7: Learning curves for (A) tanh neural network in 1 A and (B) ReLU neural network in $1 \textbf { B }$ . Both networks show good generalization with regards to the test data.
+# B ROBUSTNESS OF FINDINGS TO MI ESTIMATION METHOD AND NEURAL ACTIVATION FUNCTIONS
+This Appendix investigates the generality of the finding that compression is not observed in neural network layers with certain activation functions. Figure 1 of the main text shows example results using a binning-based MI estimator and a nonparametric KDE estimator, for both the tanh and ReLU activation functions. Here we describe the KDE MI estimator in detail, and present extended results on other datasets. We also show results for other activation functions. Finally, we provide entropy estimates based on another nonparametric estimator, the popular $\mathbf { k }$ -nearest neighbor approach of Kraskov et al. (2004). Our findings consistently show that double-saturating nonlinearities can yield compression, while single-sided nonlinearities do not.
+# B.1 KERNEL DENSITY ESTIMATION OF MI
+The KDE approach of Kolchinsky & Tracey (2017); Kolchinsky et al. (2017) estimates the mutual information between the input and the hidden layer activity by assuming that the hidden activity is distributed as a mixture of Gaussians. This assumption is well-suited to the present setting under the following interpretation: we take the input activity to be distributed as delta functions at each example in the dataset, corresponding to a uniform distribution over these specific samples. In other words, we assume that the empirical distribution of input samples is the true distribution. Next, the hidden layer activity $h$ is a deterministic function of the input. As mentioned in the main text and discussed in more detail in Appendix C, without the assumption of noise, this would have infinite mutual information with the input. We therefore assume for the purposes of analysis that Gaussian noise of variance $\sigma ^ { 2 }$ is added, that is, $T = h + \epsilon$ where $\epsilon \sim \mathcal { N } ( \bar { 0 } , \bar { \sigma ^ { 2 } } I )$ . Under these assumptions, the distribution of $T$ is genuinely a mixture of Gaussians, with a Gaussian centered on the hidden activity corresponding to each input sample. We emphasize again that the noise $\epsilon$ is added solely for the purposes of analysis, and is not present during training or testing the network. In this setting, an upper bound for the mutual information with the input is (Kolchinsky & Tracey, 2017; Kolchinsky et al., 2017)
+$$
+I ( T ; X ) \leq - { \frac { 1 } { P } } \sum _ { i } \log { \frac { 1 } { P } } \sum _ { j } \exp \left( - { \frac { 1 } { 2 } } { \frac { \vert \vert h _ { i } - h _ { j } \vert \vert _ { 2 } ^ { 2 } } { \sigma ^ { 2 } } } \right)
+$$
+where $P$ is the number of training samples and $h _ { i }$ denotes the hidden activity vector in response to input sample $i$ . Similarly, the mutual information with respect to the output can be calculated as
+$$
+\begin{array} { r c l } { I ( T ; Y ) } & { = } & { { \displaystyle H ( T ) - H ( T | Y ) } } \\ & { \leq } & { \displaystyle - \frac { 1 } { P } \sum _ { i } \log \frac { 1 } { P } \sum _ { j } \exp \left( - \frac { 1 } { 2 } \frac { \| h _ { i } - h _ { j } \| _ { 2 } ^ { 2 } } { \sigma ^ { 2 } } \right) } \\ & & { \displaystyle - \sum _ { l } ^ { L } p _ { l } \left[ - \frac { 1 } { P _ { l } } \sum _ { i , Y _ { i } = l } \log \frac { 1 } { P _ { l } } \sum _ { j , Y _ { j } = l } \exp \left( - \frac { 1 } { 2 } \frac { \| h _ { i } - h _ { j } \| _ { 2 } ^ { 2 } } { \sigma ^ { 2 } } \right) \right] } \end{array}
+$$
+where $L$ is the number of output labels, $P _ { l }$ denotes the number of data samples with output label $l$ , $p _ { l } = P _ { l } / P$ denotes the probability of output label $l$ , and the sums over $i , Y _ { i } = l$ indicate a sum over all examples with output label $l$ .
+![](images/f89ebb8650e008588c11ffbb01a6605fa94820e9aa6ba83b3408ba661f14ce97.jpg)
+Figure 8: Information plane dynamics for the network architecture and training dataset of ShwartzZiv & Tishby (2017), estimated with the nonparametric KDE method of Kolchinsky & Tracey (2017); Kolchinsky et al. (2017) and averaged over 50 repetitions. (A) tanh neural network layers show compression. (B) ReLU neural network layers show no compression. (C) The soft-sign activation function, a double-saturating nonlinearity that saturates more gently than tanh, shows modest compression. (D) The soft-plus activation function, a smoothed version of the ReLU, exhibits no compression. Hence double-saturating nonlinearities exhibit the compression effect while singlesaturating nonlinearities do not.
+Figure 8A-B shows the result of applying this MI estimation method on the dataset and network architecture of Shwartz-Ziv & Tishby (2017), with MI estimated on the full dataset and averaged over 50 repetitions. Mutual information was estimated using data samples from the test set, and we took the noise variance $\sigma ^ { 2 } = 0 . 1$ . These results look similar to the estimate derived from binning, with compression in tanh networks but no compression in ReLU networks. Relative to the binning estimate, it appears that compression is less pronounced in the KDE method.
+Figure 1C-D of the main text shows the results of this estimation technique applied to a neural network of size $7 8 4 - 1 0 2 4 - 2 0 - 2 0 - 2 0 - 1 0$ on the MNIST handwritten digit classification dataset. The network was trained using SGD with minibatches of size 128. As before, mutual information was estimated using data samples from the test set, and we took the noise variance $\sigma ^ { 2 } = 0 . 1$ . The smaller layer sizes in the top three hidden layers were selected to ensure the quality of the kernel density estimator given the amount of data in the test set, since the estimates are more accurate for smaller-dimensional data. Because of computational expense, the MNIST results are from a single training run.
+More detailed results for the MNIST dataset are provided in Figure 9 for the tanh activation function, and in Figure 10 for the ReLU activation function. In these figures, the first row shows the evolution of the cross entropy loss (on both training and testing data sets) during training. The second row shows the mutual information between input and the activity of different hidden layers, using the nonparametric KDE estimator described above. The blue region in the second row shows the range of possible MI values, ranging from the upper bound described above (Eq. 10) to the following lower
+bound (Kolchinsky & Tracey, 2017),
+$$
+\begin{array} { r c l } { { I ( T ; Y ) } } & { { \geq } } & { { \displaystyle - \frac { 1 } { P } \sum _ { i } \log \frac { 1 } { P } \sum _ { j } \exp \left( - \frac { 1 } { 2 } \frac { \left\| h _ { i } - h _ { j } \right\| _ { 2 } ^ { 2 } } { 4 \sigma ^ { 2 } } \right) } } \\ { { } } & { { } } & { { \displaystyle - \sum _ { l } ^ { L } p _ { l } \left[ - \frac { 1 } { P _ { l } } \sum _ { i , Y _ { i } = l } \log \frac { 1 } { P _ { l } } \sum _ { j , Y _ { j } = l } \exp \left( - \frac { 1 } { 2 } \frac { \left\| h _ { i } - h _ { j } \right\| _ { 2 } ^ { 2 } } { 4 \sigma ^ { 2 } } \right) \right] . } } \end{array}
+$$
+The third row shows the mutual information between input and activity of different hidden layers, estimated using the binning method (here, the activity of each neuron was discretized into bins of size 0.5). For both the second and third rows, we also plot the entropy of the inputs, $H ( X )$ , as a dashed line. $H ( X )$ is an upper bound on the mutual information $I ( X ; T )$ , and is computed using the assumption of a uniform distribution over the 10,000 testing points in the MNIST dataset, giving $H ( X ) = \log _ { 2 } { 1 0 0 0 0 }$ .
+Finally, the fourth row visualizes the dynamics of the SGD updates during training. For each layer and epoch, the green line shows the $\ell _ { 2 }$ norm of the weights. We also compute the vector of mean updates across SGD minibatches (this vector has one dimension for each weight parameter), as well as the vector of the standard deviation of the updates across SGD minibatches. The $\ell _ { 2 }$ norm of the mean update vector is shown in blue, and the $\ell _ { 2 }$ norm of the standard deviation vector is shown in orange. The gradient SNR, computed as the ratio of the norm of the mean vector to the norm of the standard deviation vector, is shown in red. For both the tanh and ReLU networks, the gradient SNR shows a phase transition during training, and the norm of the weights in each layer increases. Importantly, this phase transition occurs despite a lack of compression in the ReLU network, indicating that noise in SGD updates does not yield compression in this setting.
+![](images/cc8295ba9fbcdf4553fd120007b118b080b408782fb4c7b756ca91d6a943022a.jpg)
+Figure 9: Detailed tanh activation function results on MNIST. Row 1: Loss over training. Row 2: Upper and lower bounds for the mutual information $I ( X ; T )$ between the input $( X )$ and each layer’s activity $( T )$ , using the nonparametric KDE estimator (Kolchinsky & Tracey, 2017; Kolchinsky et al., 2017). Dotted line indicates $H ( X ) = \log _ { 2 } { 1 0 0 0 0 }$ , the entropy of a uniform distribution over 10,000 testing samples. Row 3: Binning-based estimate of the mutual information $I ( X ; T )$ , with each neuron’s activity discretized using a bin size of 0.5. Row 4: Gradient SNR and weight norm dynamics. The gradient SNR shows a phase transition during training, and the norm of the weights in each layer increases.
+![](images/6b5d4819da8c875280c0636b21620dd6a2ea91f5c10ae7ab67fc192db6fbb0d4.jpg)
+Figure 10: Detailed ReLU activation function results on MNIST. Row 1: Loss over training. Row 2: Upper and lower bounds for the mutual information $I ( X ; T )$ between the input $( X )$ and each layer’s activity $( T )$ , using the nonparametric KDE estimator (Kolchinsky & Tracey, 2017; Kolchinsky et al., 2017). Dotted line indicates $H ( X ) = \log _ { 2 } { 1 0 0 0 0 }$ , the entropy of a uniform distribution over 10,000 testing samples. Row 3: Binning-based estimate of the mutual information $I ( X ; T )$ , with each neuron’s activity discretized using a bin size of 0.5. Row 4: Gradient SNR and weight norm dynamics. The gradient SNR shows a phase transition during training, and the norm of the weights in each layer increases. Importantly, this phase transition occurs despite a lack of compression in the ReLU network, indicating that noise in SGD updates does not yield compression in this setting.
+![](images/6c1149a7423a2791869e903f30e00c9268d38906bea86674e43fe62d02d3186f.jpg)
+Figure 11: Alternative activation functions.
+# B.2 OTHER ACTIVATION FUNCTIONS
+Next, in Fig. 8C-D, we show results from the kernel MI estimator from two additional nonlinear activation functions, the softsign function
+$$
+f ( x ) = { \frac { x } { 1 + | x | } } ,
+$$
+and the softplus function
+$$
+f ( x ) = \ln ( 1 + e ^ { x } ) .
+$$
+These functions are plotted next to tanh and ReLU in Fig. 11. The softsign function is similar to tanh but saturates more slowly, and yields less compression than tanh. The softplus function is a smoothed version of the ReLU, and yields similar dynamics with no compression. Because softplus never saturates fully to zero, it retains more information with respect to the input than ReLUs in general.
+# B.3 KRASKOV ESTIMATOR
+We additionally investigated the widely-used nonparametric MI estimator of Kraskov et al. (2004). This estimator uses nearest neighbor distances between samples to compute an estimate of the entropy of a continuous random variable. Here we focused for simplicity only on the compression phenomenon in the mutual information between the input and hidden layer activity, leaving aside the information with respect to the output (as this is not relevant to the compression phenomenon). Again, without additional noise assumptions, the MI between the hidden representation and the input would be infinite because the mapping is deterministic. Rather than make specific noise assumptions, we instead use the Kraskov method to estimate the entropy of the hidden representations $T$ . Note that the entropy of $T$ is the mutual information up to an unknown constant so long as the noise assumption is homoscedastic, that is, $T = h + Z$ where the random variable $Z$ is independent of $X$ . To see this, note that
+$$
+\begin{array} { l l l } { { I ( T ; X ) } } & { { = } } & { { H ( T ) - H ( T | X ) } } \\ { { } } & { { = } } & { { H ( T ) - H ( Z ) } } \\ { { } } & { { = } } & { { H ( T ) - c } } \end{array}
+$$
+where the constant $c = H ( Z )$ . Hence observing compression in the layer entropy $H ( T )$ is enough to establish that compression occurs in the mutual information.
+The Kraskov estimate is given by
+$$
+\frac { d } { P } \sum _ { i = 1 } ^ { P } \log ( r _ { i } + \epsilon ) + \frac { d } { 2 } \log ( \pi ) - \log \Gamma ( d / 2 + 1 ) + \psi ( P ) - \psi ( k )
+$$
+where $d$ is the dimension of the hidden representation, $P$ is the number of samples, $r _ { i }$ is the distance to the $k$ -th nearest neighbor of sample $i$ , $\epsilon$ is a small constant for numerical stability, $\Gamma ( \cdot )$ is the
+Gamma function, and $\psi ( \cdot )$ is the digamma function. Here the parameter $\epsilon$ prevents infinite terms when the nearest neighbor distance ri = 0 for some sample. We took  = 10−16.
+Figure 12 shows the entropy over training for tanh and ReLU networks trained on the dataset of and with the network architecture in Shwartz-Ziv & Tishby (2017), averaged over 50 repeats. In these experiments, we used $k = 2$ . Compression would correspond to decreasing entropy over the course of training, while a lack of compression would correspond to increasing entropy. Several tanh layers exhibit compression, while the ReLU layers do not. Hence qualitatively, the Kraskov estimator returns similar results to the binning and KDE strategies.
+![](images/d35dbe7c6e40d832895cc4e561a74abc5f2ec6a2b6e899768df8078d5351eb42.jpg)
+Figure 12: Entropy dynamics over training for the network architecture and training dataset of Shwartz-Ziv & Tishby (2017), estimated with the nonparametric $\mathbf { k }$ -nearest-neighbor-based method of Kraskov et al. (2004). Here the $\mathbf { X } ^ { } -$ -axis is epochs of training time, and the y-axis plots the entropy of the hidden representation, as calculated using nearest-neighbor distances. Note that in this setting, if $T$ is considered to be the hidden activity plus independent noise, the entropy is equal to the mutual information up to a constant (see derivation in text). Layers 0-4 correspond to the hidden layers of size 10-7-5-4-3. (A) tanh neural network layers can show compression over the course of training. (B) ReLU neural network layers show no compression.
+# C NOISE ASSUMPTIONS AND DISCRETE VS CONTINUOUS ENTROPY
+A recurring theme in the results reported in this paper is the necessity of noise assumptions to yield a nontrivial information theoretic analysis. Here we give an extended discussion of this phenomenon, and of issues relating to discrete entropy as opposed to continuous (differential) entropy.
+The activity of a neural network is often a continuous deterministic function of its input. That is, in response to an input $X$ , a specific hidden layer might produce activity $h = f ( X )$ for some function $f$ . The mutual information between $h$ and $X$ is given by
+$$
+\begin{array} { l l l } { { I ( h ; X ) } } & { { = } } & { { H ( h ) - H ( h | X ) . } } \end{array}
+$$
+If $h$ were a discrete variable, then the entropy would be given by
+$$
+H ( h ) = - \sum _ { i = 1 } ^ { N } p _ { i } \log p _ { i }
+$$
+where $p _ { i }$ is the probability of the discrete symbol $i$ , as mentioned in the main text. Then $H ( h | X ) = 0$ because the mapping is deterministic and we have $I ( h ; X ) = H ( h )$ .
+However $h$ is typically continuous. The continuous entropy, defined for a continuous random variable $Z$ with density $p _ { Z }$ by analogy to Eqn. (18) as
+$$
+H ( Z ) = - \int p _ { Z } ( z ) \log p _ { Z } ( z ) d z ,
+$$
+can be negative and possibly infinite. In particular, note that if $p _ { Z }$ is a delta function, then $H ( Z ) =$ $- \infty$ . The mutual information between hidden layer activity $h$ and the input $X$ for continuous $h , X$ is
+$$
+I ( h ; X ) = H ( h ) - H ( h | X ) .
+$$
+Now $H ( h | X ) = - \infty$ since given the input $X$ , the hidden activity $h$ is distributed as a delta function at $f ( X )$ . The mutual information is thus generally infinite, so long as the hidden layer activity has finite entropy $H ( h )$ is finite).
+![](images/db291cd24878a5a577925a9f57c17f1e80ff0653e022e4f8350fe21e76b70f02.jpg)
+Figure 13: Effect of binning strategy on minimal three neuron model. Mutual information for the simple three neuron model shown in Fig. 2 with bin edges $b _ { i } \in \mathrm { t a n h } ( \operatorname* { l i n s p a c e } ( - 5 0 , 5 0 , N ) )$ . In contrast to linear binning, the mutual information continues to increase as weights grow.
+To yield a finite mutual information, some noise in the mapping is required such that $H ( h | X )$ remains finite. A common choice (and one adopted here for the linear network, the nonparametric kernel density estimator, and the $\mathbf { k }$ -nearest neighbor estimator) is to analyze a new variable with additive noise, $T = h + Z$ , where $Z$ is a random variable independent of $X$ . Then $H ( T | X ) = H ( Z )$ which allows the overall information $I ( T ; X ) = H ( T ) - H ( Z )$ to remain finite. This noise assumption is not present in the actual neural networks either during training or testing, and is made solely for the purpose of calculating the mutual information.
+Another strategy is to partition the continuous variable $h$ into a discrete variable $T$ , for instance by binning the values (the approach taken in Shwartz-Ziv & Tishby (2017)). This allows use of the discrete entropy, which remains finite. Again, however, in practice the network does not operate on the binned variables $T$ but on the continuous variables $h$ , and the binning is solely for the purpose of calculating the mutual information. Moreover, there are many possible binning strategies, which yield different discrete random variables, and different mutual information with respect to the input. The choice of binning strategy is an assumption analogous to choosing a type of noise to add to the representation in the continuous case: because there is in fact no binning in the operation of the network, there is no clear choice for binning methodology. The strategy we use in binning-based experiments reported here is the following: for bounded activations like the tanh activation, we use evenly spaced bins between the minimum and maximum limits of the function. For unbounded activations like ReLU, we first train the network completely; next identify the minimum and maximum hidden activation over all units and all training epochs; and finally bin into equally spaced bins between these minimum and maximum values. We note that this procedure places no restriction on the magnitude that the unbounded activation function can take during training, and yields the same MI estimate as using infinite equally spaced bins (because bins for activities larger than the maximum are never seen during training).
+As an example of another binning strategy that can yield markedly different results, we consider evenly spaced bins in a neuron’s net input, rather than its activity. That is, instead of evenly spaced bins in the neural activity, we determine the bin edges by mapping a set of evenly spaced values through the neural nonlinearity. For tanh, for instance, this spaces bins more tightly in the saturation region as compared to the linear region. Figure 13 shows the results of applying this binning strategy to the minimal three neuron model with tanh activations. This binning scheme captures more information as the weights of the network grow larger. Figure 14 shows information plane dynamics for this binning structure. The tanh network no longer exhibits compression. (We note that the broken DPI in this example is an artifact of performing binning only for analysis, as discussed below).
+![](images/12c7d31d8a03d347bac1cbe449138df659dc492793257cc0c61d13e2608ea4fa.jpg)
+Figure 14: Effect of binning strategy on information plane dynamics. Results for the same tanh network and training regime as 1A, but with bin edges $b _ { i } \in$ tanh(linspace $( - 5 0 , 5 0 , N )$ ). Measured with this binning structure, there is no compression in most layers.
+Any implementation of a neural network on digital hardware is ultimately of finite precision, and hence is a binned, discrete representation. However, it is a very high resolution binning compared to that used here or by Shwartz-Ziv & Tishby (2017): single precision would correspond to using roughly $2 ^ { 3 2 }$ bins to discretize each hidden unit’s activity, as compared to the 30-100 used here. If the binning is fine-grained enough that each input $X$ yields a different binned activity pattern $h$ , then $H ( h ) \stackrel { - } { = } \log ( P )$ where $P$ is the number of examples in the dataset, and there will be little to no change in information during training. As an example, we show in Fig. 15 the result of binning at full machine precision.
+Finally, we note two consequences of the assumption of noise/binning for the purposes of analysis. First, this means that the data processing inequality (DPI) does not apply to the noisy/binned mutual information estimates. The DPI states that information can only be destroyed through successive transformations, that is, if $X  h _ { 1 }  h _ { 2 }$ form a Markov chain, then $I ( X ; \bar { h _ { 1 } } ) \geq I ( \bar { X ; } h _ { 2 } )$ (see, eg, Tishby & Zaslavsky (2015)). Because noise is added only for the purpose of analysis, however, this does not apply here. In particular, for the DPI to apply, the noise added at lower layers would have to propagate through the network to higher layers. That is, if the transformation from hidden layer 1 to hidden layer 2 is $h _ { 2 } = f ( h _ { 1 } )$ and $T _ { 1 } = h _ { 1 } + Z _ { 1 }$ is the hidden layer activity after adding noise, then the DPI would hold for the variable $\tilde { T } _ { 2 } = f ( T _ { 1 } ) + Z _ { 2 } = f ( h _ { 1 } + Z _ { 1 } ) + Z _ { 2 }$ , not the quantity
+![](images/e2187148ec2ae4d72ade118f8c38e42693609c24edb83c7050bd489b4d6120ca.jpg)
+Figure 15: Effect of binning at full machine precision. (A) ReLU network. (B) tanh network. Information in most layers stays pinned to $\log _ { 2 } ( P ) = 1 2$ . Compression is only observed in the highest and smallest layers near the very end of training, when the saturation of tanh is strong enough to saturate machine precision.
+$T _ { 2 } = h _ { 2 } + Z _ { 2 } = f ( h _ { 1 } ) + Z _ { 2 }$ used in the analysis. Said another way, the Markov chain for $T _ { 2 }$ is $X  h _ { 1 }  h _ { 2 }  T _ { 2 }$ , not $X  h _ { 1 }  T _ { 1 }  T _ { 2 }$ , so the DPI states only that $I ( X ; h _ { 1 } ) \ge I ( X ; T _ { 2 } )$ .
+A second consequence of the noise assumption is the fact that the mutual information is no longer invariant to invertible transformations of the hidden activity $h$ . A potentially attractive feature of a theory based on mutual information is that it can allow for comparisons between different architectures: mutual information is invariant to any invertible transformation of the variables, so two hidden representations could be very different in detail but yield identical mutual information with respect to the input. However, once noise is added to a hidden representation, this is no longer the case: the variable $T = h + Z$ is not invariant to reparametrizations of $h$ . As a simple example, consider a minimal linear network with scalar weights $w _ { 1 }$ and $w _ { 2 }$ that computes the output ${ \hat { y } } = w _ { 2 } w _ { 1 } X$ . The hidden activity is $h = w _ { 1 } X$ . Now consider the family of networks in which we scale down $w _ { 1 }$ and scale up $w _ { 2 }$ by a factor $c \neq 0$ , that is, these networks have weights $\tilde { w } _ { 1 } = w _ { 1 } / c$ and $\tilde { w } _ { 2 } = c w _ { 2 }$ , yielding the exact same input-output map $\hat { y } = \tilde { w } _ { 2 } \tilde { w } _ { 1 } X = c w _ { 2 } ( w _ { 1 } / c ) X = w _ { 2 } w _ { 1 } X$ . Because they compute the same function, they necessarily generalize identically. However after introducing the noise assumption the mutual information is
+$$
+\begin{array} { l l l } { { I ( T ; X ) } } & { { = } } & { { \log \left( w _ { 1 } ^ { 2 } / c ^ { 2 } + \sigma _ { M I } ^ { 2 } \right) - \log \left( \sigma _ { M I } ^ { 2 } \right) } } \end{array}
+$$
+where we have taken the setting in Section 3 in which $X$ is normal Gaussian, and independent Gaussian noise of variance $\sigma _ { M I } ^ { 2 }$ is added for the purpose of MI computation. Clearly, the mutual information is now dependent on the scaling $c$ of the internal layer, even though this is an invertible linear transformation of the representation. Moreover, this shows that networks which generalize identically can nevertheless have very different mutual information with respect to the input when it is measured in this way.
+# D WEIGHT NORMS OVER TRAINING
+Our argument relating neural saturation to compression in mutual information relies on the notion that in typical training regimes, weights begin small and increase in size over the course of training. We note that this is a virtual necessity for a nonlinearity like tanh, which is linear around the origin: when initialized with small weights, the activity of a tanh network will be in this linear regime and the network can only compute a linear function of its input. Hence a real world nonlinear task can only be learned by increasing the norm of the weights so as to engage the tanh nonlinearity on some examples. This point can also be appreciated from norm-based capacity bounds on neural networks, which show that, for instance, the Rademacher complexity of a neural network with small weights must be low (Bartlett & Mendelson, 2002; Neyshabur et al., 2015). Finally, as an empirical matter, the networks trained in this paper do in fact increase the norm of their weights over the course of training, as shown by the green lines in Figure 20 for tanh and ReLU networks in the training setting of Shwartz-Ziv & Tishby (2017); Figures 9 and 10 for the MNIST networks; and Figure 21 for a linear network.
+# E HISTOGRAMS OF NEURAL ACTIVATIONS
+Supplementary Figures 16 and 17 show histograms of neural activities over the course of training in tanh and ReLU networks respectively.
+![](images/8bbc07b14318e40e4fb277a40f930af47fb6bee0d59af7119d1a65cf2931ed0c.jpg)
+Figure 16: Histogram of neural activities in a tanh network during training. The final three layers eventually saturate in the top and bottom bins corresponding to the saturation limits of the tanh activation function, explaining the compression observed in tanh. x-axis: training time in epochs. y-axis: Hidden activity bin values from lowest to highest. Colormap: density of hidden layer activities across all input examples.
+![](images/5f209aa29a04ff3253199dfe5ae51898ad1ec81c4d32a5a079b30809f462dc92.jpg)
+Figure 17: Histogram of neural activities in a ReLU network during training. ReLU layers 1-5 have a roughly constant fraction of activities at zero, corresponding to instances where the ReLU is off; the nonzero activities disperse over the course of training without bound, yielding higher entropy distributions. The sigmoid output layer 6 converges to its saturation limits, and is the only layer that compresses during training (c.f. Fig. 1B). $\mathbf { X }$ -axis: training time in epochs. y-axis: Hidden activity value. Colormap: density of hidden layer activities across all input examples.
+# F INFORMATION PLANE DYNAMICS IN DEEPER LINEAR NETWORKS
+Supplementary Figure 18 shows information plane dynamics for a deep neural network with five hidden layers each containing 50 hidden units.
+![](images/cba1fe5de44777e480b97d883918063f6902a3728f471a1abd20879b524f24ab.jpg)
+Figure 18: Information plane dynamics in a deep linear neural network. (A) Train and test error during learning. (B) Information plane dynamics. No compression is visible.
+# G LINEAR MUTUAL INFORMATION CALCULATION
+For the linear setting considered here, the mutual information between a hidden representation $T$ and the output $Y$ may be calculated using the relations
+$$
+\begin{array} { c } { { { \cal H } ( Y ) = \displaystyle \frac { N _ { o } } { 2 } \log ( 2 \pi e ) + \displaystyle \frac { 1 } { 2 } \log | W _ { o } W _ { o } ^ { T } + \sigma _ { o } ^ { 2 } I _ { N _ { o } } | , } } \\ { { { \cal H } ( T ) = \displaystyle \frac { N _ { h } } { 2 } \log ( 2 \pi e ) + \displaystyle \frac { 1 } { 2 } \log | \bar { W } \bar { W } ^ { T } + \sigma _ { M I } ^ { 2 } I _ { N _ { h } } | , } } \\ { { { \cal H } ( Y ; T ) = \displaystyle \frac { N _ { o } + N _ { h } } { 2 } \log ( 2 \pi e ) + \displaystyle \frac { 1 } { 2 } \log | \bar { W } \bar { W } ^ { T } + \sigma _ { M I } ^ { 2 } I _ { N _ { h } } \quad \quad \bar { W } W _ { o } ^ { T } ,  } } \\ { { { \cal  I } ( Y ; T ) = { \cal H } ( Y ) + { \cal H } ( T ) - { \cal H } ( Y ; T ) . } } \end{array}
+$$
+# H STOCHASTIC VS BATCH TRAINING
+![](images/547f68d75db31823ea92f3a340ffd0af1f2555a96902c610b5d4b54c25c651f4.jpg)
+Figure 19 shows information plane dynamics for stochastic and batch gradient descent learning in a linear network. Randomness in the training process does not dramatically alter the information plane dynamics.
+Figure 19: Effect of stochastic training in linear networks. (A) Information plane dynamics for stochastic gradient descent in a linear network (same setting as Fig. 4). (B) Information plane dynamics for batch gradient descent.
+# I GRADIENT SNR PHASE TRANSITION
+The proposed mechanism of compression in Shwartz-Ziv & Tishby (2017) is noise arising from stochastic gradient descent training. The results in Section 4 of the main text show that compression still occurs under batch gradient descent learning, suggesting that in fact noise in the gradient updates is not the cause of compression. Here we investigate a related claim, namely that during training, networks switch between two phases. These phases are defined by the ratio of the mean of the gradient to the standard deviation of the gradient across training examples, called the gradient signal-to-noise ratio. In the first “drift” phase, the SNR is high, while in the second “diffusion” phase the SNR is low. Shwartz-Ziv & Tishby (2017) hypothesize that the drift phase corresponds to movement toward the minimum with no compression, while the diffusion phase corresponds to a constrained diffusion in weight configurations that attain the optimal loss, during which representations compress. However, two phases of gradient descent have been described more generally, sometimes known as the transient and stochastic phases or search and convergence phases (Murata, 1998; Chee & Toulis, 2017), suggesting that these phases might not be related specifically to compression behavior.
+In Fig. 20 we plot the gradient SNR over the course of training for the tanh and ReLU networks in the standard setting of Shwartz-Ziv & Tishby (2017). In particular, for each layer $l$ we calculate the mean and standard deviation as
+$$
+\begin{array} { r c l } { { m _ { l } } } & { { = } } & { { \displaystyle \left\| \left. \frac { \partial E } { \partial W _ { l } } \right. \right\| _ { F } } } \\ { { s _ { l } } } & { { = } } & { { \displaystyle \left\| \mathrm { S T D } \left( \frac { \partial E } { \partial W _ { l } } \right) \right\| _ { F } } } \end{array}
+$$
+where $\langle \cdot \rangle$ denotes the mean and $S T D ( \cdot )$ denotes the element-wise standard deviation across all training samples, and $\left\| \cdot \right\| _ { F }$ denotes the Frobenius norm. The gradient SNR is then the ratio $m _ { l } / s _ { l }$ We additionally plot the norm of the weights $\| W _ { l } \| _ { F }$ over the course of training.
+Both tanh and ReLU networks yield a similar qualitative pattern, with SNR undergoing a step-like transition to a lower value during training. Figures 9 and 10, fourth row, show similar plots for MNIST-trained networks. Again, SNR undergoes a transition from high to low over training. Hence the two phase nature of gradient descent appears to hold across the settings that we examine here. Crucially, this finding shows that the SNR transition is not related to the compression phenomenon because ReLU networks, which show the gradient SNR phase transition, do not compress.
+Finally, to show the generality of the two-phase gradient SNR behavior and its independence from compression, we develop a minimal model of this phenomenon in a three neuron linear network. We consider the student-teacher setting of Fig. 3 but with $N _ { i } = N _ { h } = 1$ , such that the input and hidden layers have just a single neuron (as in the setting of Fig. 2). Here, with just a single hidden neuron, clearly there can be no compression so long as the first layer weight increases over the course of training. Figure 21AC shows that even in this simple setting, the SNR shows the phase transition but the weight norm increases over training. Hence again, the two phases of the gradient are present even though there is no compression. To intuitively understand the source of this behavior, note that the weights are initialized to be small and hence early in learning all must be increased in magnitude, yielding a consistent mean gradient. Once the network reaches the vicinity of the minimum, the mean weight change across all samples by definition goes to zero. The standard deviation remains finite, however, because on some specific examples error could be improved by increasing or decreasing the weights–even though across the whole dataset the mean error has been minimized.
+Hence overall, our results show that a two-phase structure in the gradient SNR occurs in all settings we consider, even though compression occurs only in a subset. The gradient SNR behavior is therefore not causally related to compression dynamics, consistent with the view that saturating nonlinearities are the primary source of compression.
+![](images/fb8ca7128e6032058471eb13f01a0a5dba496e6d717ed509113be8ea4d25e22c.jpg)
+Figure 20: Gradient SNR phase transition. (A) tanh networks trained in the standard setting of Shwartz-Ziv & Tishby (2017) show a phase transition in every layer. (B) ReLU networks also show a phase transition in every layer, despite exhibiting no compression.
+![](images/a9a3a5f252886534d662bb947459c62130ced799c330bd5e27586470c9beb20b.jpg)
+Figure 21: Minimal model exhibiting gradient SNR phase transition. Here a three neuron linear network (architecture $1 - 1 - 1 )$ learns to approximate a teacher. Other parameters are teacher $S N R = 1$ , number of training samples $P = 1 0 0$ , learning rate .001. Left column: (A) The loss over training with SGD (minibatch size 1). (C) The resulting gradient SNR dynamics. Right column: (B) The loss over training with BGD. (D) The resulting gradient SNR dynamics averaging over all training samples (not minibatches, see text).