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Title: Sparse Attentive Backtracking: Long-Range Credit Assignment in Recurrent Networks

Paper content:
A major drawback of backpropagation through time (BPTT) is the difficulty of learning long-term dependencies, coming from having to propagate credit information backwards through every single step of the forward computation.
This makes BPTT both computationally impractical and biologically implausible.
For this reason, full backpropagation through time is rarely used on long sequences, and truncated backpropagation through time is used as a heuristic.
However, this usually leads to biased estimates of the gradient in which longer term dependencies are ignored.
Addressing this issue, we propose an alternative algorithm, Sparse Attentive Backtracking, which might also be related to principles used by brains to learn long-term dependencies.
Sparse Attentive Backtracking learns an attention mechanism over the hidden states of the past and selectively backpropagates through paths with high attention weights.
This allows the model to learn long term dependencies while only backtracking for a small number of time steps, not just from the recent past but also from attended relevant past states.
Recurrent Neural Networks (RNNs) are state-of-the-art for many machine learning sequence processing tasks.
Examples where models based on RNNs shine include speech recognition BID21 BID3 , image captioning BID32 BID29 BID17 , machine translation BID1 BID26 BID18 , and speech synthesis BID20 .
It is common practice to train these models using backpropagation through time (BPTT), wherein the network states are unrolled in time and gradients are backpropagated through the unrolled graph.
Since the parameters of an RNN are shared across the different time steps, BPTT is more prone to vanishing and exploding gradients (Hochreiter, 1991; BID2 BID11 than equivalent deep feedforward networks with as many stages.
This makes credit assignment particularly difficult for events that have occurred many time steps in the past, and thus makes it challenging in practice to capture long-term dependencies in the data (Hochreiter, 1991; BID2 .
Having to wait for the end of the sequence in order to compute gradients is neither practical for machines nor animals when the dependencies extend over very long timescales.
Training is slowed down considerably by long waiting times, as the rate of convergence crucially depends on how often parameters can be updated.In practice, proper long-term credit assignment in RNNs is very inconvenient, and it is common practice to employ truncated versions of BPTT for long sequences BID23 BID24 .
In truncated BPTT (TBPTT), gradients are backpropagated only for a fixed and limited number of time steps and parameters are updated after each such subsequence.
Truncation is often motivated by computational concerns: memory, computation time and the advantage of faster learning obtained when making more frequent updates of the parameters rather than having to wait for the end of the sequence.
However, it makes capturing correlations across distant states even harder.Regular RNNs are parametric: their hidden state vector has a fixed size.
We believe that this is a critical element in the classical analysis of the difficulty of learning long-term dependencies BID2 .
Indeed, the fixed state dimension becomes a bottleneck through which information has to flow, both forward and backward.We thus propose a semi-parametric RNN, where the next state is potentially conditioned on all the previous states of the RNN, making it possible-thanks to attention-to jump through any distance through time.
We distinguish three types of states in our proposed semi-parametric RNN:• The fixed-size hidden state h (t) , the conventional state of an RNN model at time t;• The monotonically-growing macrostate M = {m (1) , . . . , m
(s) }, the array of all past microstates, which plays the role of a random-access memory;• And the fixed-size microstate m (i) , which is the ith hidden state (one of the h (t) ) that was chosen for inclusion within the macrostate M. There are as many hidden states as there are timesteps in the sequence being analyzed by the RNN.
A subset of them will become microstates, and this subset is called the macrostate.The computation of the next hidden state h (t+1) is based on the whole macrostate M, in addition to the external input x (t) .
The macrostate being variable-length, we must devise a special mechanism to read from this ever-growing array.
As a key component of our model, we propose to use an attention mechanism over the microstate elements of the macrostate.The attention mechanism in the above setting may be regarded as providing adaptive, dynamic skip connections: any past microstate can be linked, via a dynamic decision, to the current hidden state.
Skip connections allow information to propagate over very long sequences.
Such architectures should naturally make it easier to learn long-term dependencies.
We name our algorithm sparse attentive backtracking (SAB).
SAB is especially well-suited to sequences in which two parts of a task are closely related yet occur very far apart in time.Inference in SAB involves examining the macrostate and selecting some of its microstates.
Ideally, SAB will not select all microstates, instead attending only to the most salient or relevant ones (e.g., emotionally loaded, in animals).
The attention mechanism will select a number of relevant microstates to be incorporated into the hidden state.
During training, local backpropagation of gradients happens in a short window of time around the selected microstates only.
This allows for the updates to be asynchronous with respect to the time steps we attend to, and credit assignment takes place more globally in the proposed algorithm.With the proposed framework for SAB, we present the following contributions:• A principled way of doing sparse credit assignment, based on a semi-parametric RNN.•
A novel way of mitigating exploding and vanishing gradients, based on reducing the number of steps that need to be backtracked through temporal skip connections.•
Competitive results compared to full backpropagation through time (BPTT), and much better results as compared to Truncated Backpropagation through time, with significantly shorter truncation windows in our model. Mechanisms
such as SAB may also be biologically plausible. Imagine having
taken a wrong turn on a roadtrip and finding out about it several miles later. Our mental focus
would most likely shift directly to the location in time and space where we had made the wrong decision, without replaying in reverse the detailed sequence of experienced traffic and landscape impressions. Neurophysiological
findings support the existence of such attention mechanisms and their involvement in credit assignment and learning in biological systems. In particular, hippocampal
recordings in rats indicate that brief sequences of prior experience are replayed both in the awake resting state and during sleep, both of which conditions are linked to memory consolidation and learning BID7 BID6 BID8 . Moreover, it has been observed
that these replay events are modulated by the reward an animal does or does not receive at the end of a task in the sense that they are more pronounced in the presence of a reward signal and less pronounced or absent in the absence of a reward signal BID0 . Thus, the mental look back into
the past seems to occur exactly when credit assignment is to be performed.2 RELATED WORK 2.1 TRUNCATED BACKPROPAGATION
THROUGH TIME When training on very long sequences, full backpropagation through time becomes computationally expensive and considerably slows down training by forcing the learner to wait for the end of each (possibly very long sequence) before making a parameter update. A common heuristic is to backpropagate the
loss of a particular time step through only a limited number of time steps, and hence truncate the backpropagation computation graph BID30 . While truncated backpropagation through time
is heavily used in practice, its inability to perform credit assignment over longer sequences is a limiting factor for this algorithm, resulting in failure cases even in simple tasks, such as the Copying Memory and Adding task in BID12 .
Improving the modeling of long-term dependencies is a central challenge in sequence modeling, and the exact gradient computation by BPTT is not biologically plausible as well as inconvenient computationally for realistic applications.
Because of this, the most widely used algorithm for training recurrent neural networks on long sequences is truncated backpropagation through time, which is known to produced biased estimates of the gradient , focusing on short-term dependencies.
We have proposed Sparse Attentive Backtracking, a new biologically motivated algorithm which aims to combine the strengths of full backpropagation through time and truncated backpropagation through time.
It does so by only backpropagating gradients through paths selected by its attention mechanism.
This allows the RNN to learn long-term dependencies, as with full backpropagation through time, while still allowing it to only backtrack for a few steps, as with truncated backpropagation through time, thus making it possible to update weights as frequently as needed rather than having to wait for the end of very long sequences.

assistant: text: Towards Efficient Credit Assignment in Recurrent Networks without Backpropagation Through Time