#

PonderNet: Learning to Ponder

This is a PyTorch implementation of the paper PonderNet: Learning to Ponder.

PonderNet adapts the computation based on the input. It changes the number of steps to take on a recurrent network based on the input. PonderNet learns this with end-to-end gradient descent.

PonderNet has a step function of the form

$\overset{y}{^}_{n}, h_{n + 1}, λ_{n} = s (x, h_{n})$

where $x$ is the input, $h_{n}$ is the state, $\overset{y}{^}_{n}$ is the prediction at step $n$ , and $λ_{n}$ is the probability of halting (stopping) at current step.

$s$ can be any neural network (e.g. LSTM, MLP, GRU, Attention layer).

The unconditioned probability of halting at step $n$ is then,

$p_{n} = λ_{n} j = 1 \prod n - 1 (1 - λ_{j})$

That is the probability of not being halted at any of the previous steps and halting at step $n$ .

During inference, we halt by sampling based on the halting probability $λ_{n}$ and get the prediction at the halting layer $\overset{y}{^}_{n}$ as the final output.

During training, we get the predictions from all the layers and calculate the losses for each of them. And then take the weighted average of the losses based on the probabilities of getting halted at each layer $p_{n}$ .

The step function is applied to a maximum number of steps donated by $N$ .

The overall loss of PonderNet is

L L_{R ec} L_{R e g} = L_{R ec} + β L_{R e g} = n = 1 \sum N p_{n} L (y, \overset{y}{^}_{n}) = K L (p_{n} ∥ p_{G} (λ_{p}))

$L$ is the normal loss function between target $y$ and prediction $\overset{y}{^}_{n}$ .

$K L$ is the Kullback–Leibler divergence.

$p_{G}$ is the Geometric distribution parameterized by $λ_{p}$ . $λ_{p}$ has nothing to do with $λ_{n}$ ; we are just sticking to same notation as the paper. $P r_{p_{G} (λ_{p})} (X = k) = (1 - λ_{p})^{k} λ_{p}$ .

The regularization loss biases the network towards taking $\frac{1}{λ _{p}}$ steps and incentivizes non-zero probabilities for all steps; i.e. promotes exploration.

Here is the training code experiment.py to train a PonderNet on Parity Task.

65from typing import Tuple
66
67import torch
68from torch import nn
69
70from labml_helpers.module import Module

#

PonderNet with GRU for Parity Task

This is a simple model that uses a GRU Cell as the step function.

This model is for the Parity Task where the input is a vector of n_elems . Each element of the vector is either 0 , 1 or -1 and the output is the parity - a binary value that is true if the number of 1 s is odd and false otherwise.

The prediction of the model is the log probability of the parity being $1$ .

73class ParityPonderGRU(Module):

#

n_elems is the number of elements in the input vector
n_hidden is the state vector size of the GRU
max_steps is the maximum number of steps $N$

87    def __init__(self, n_elems: int, n_hidden: int, max_steps: int):

#

93        super().__init__()
94
95        self.max_steps = max_steps
96        self.n_hidden = n_hidden

#

GRU $h_{n + 1} = s_{h} (x, h_{n})$

100        self.gru = nn.GRUCell(n_elems, n_hidden)

#

$\overset{y}{^}_{n} = s_{y} (h_{n})$ We could use a layer that takes the concatenation of $h$ and $x$ as input but we went with this for simplicity.

104        self.output_layer = nn.Linear(n_hidden, 1)

#

$λ_{n} = s_{λ} (h_{n})$

106        self.lambda_layer = nn.Linear(n_hidden, 1)
107        self.lambda_prob = nn.Sigmoid()

#

An option to set during inference so that computation is actually halted at inference time

109        self.is_halt = False

#

x is the input of shape [batch_size, n_elems]

This outputs a tuple of four tensors:

1. $p_{1} \dots p_{N}$ in a tensor of shape [N, batch_size] 2. $\overset{y}{^}_{1} \dots \overset{y}{^}_{N}$ in a tensor of shape [N, batch_size] - the log probabilities of the parity being $1$ 3. $p_{m}$ of shape [batch_size] 4. $\overset{y}{^}_{m}$ of shape [batch_size] where the computation was halted at step $m$

111    def forward(self, x: torch.Tensor) -> Tuple[torch.Tensor, torch.Tensor, torch.Tensor, torch.Tensor]:

#

124        batch_size = x.shape[0]

#

We get initial state $h_{1} = s_{h} (x)$

127        h = x.new_zeros((x.shape[0], self.n_hidden))
128        h = self.gru(x, h)

#

Lists to store $p_{1} \dots p_{N}$ and $\overset{y}{^}_{1} \dots \overset{y}{^}_{N}$

131        p = []
132        y = []

#

$\prod_{j = 1}^{n - 1} (1 - λ_{j})$

134        un_halted_prob = h.new_ones((batch_size,))

#

A vector to maintain which samples has halted computation

137        halted = h.new_zeros((batch_size,))

#

$p_{m}$ and $\overset{y}{^}_{m}$ where the computation was halted at step $m$

139        p_m = h.new_zeros((batch_size,))
140        y_m = h.new_zeros((batch_size,))

#

Iterate for $N$ steps

143        for n in range(1, self.max_steps + 1):

#

The halting probability $λ_{N} = 1$ for the last step

145            if n == self.max_steps:
146                lambda_n = h.new_ones(h.shape[0])

#

$λ_{n} = s_{λ} (h_{n})$

148            else:
149                lambda_n = self.lambda_prob(self.lambda_layer(h))[:, 0]

#

$\overset{y}{^}_{n} = s_{y} (h_{n})$

151            y_n = self.output_layer(h)[:, 0]

#

$p_{n} = λ_{n} j = 1 \prod n - 1 (1 - λ_{j})$

154            p_n = un_halted_prob * lambda_n

#

Update $\prod_{j = 1}^{n - 1} (1 - λ_{j})$

156            un_halted_prob = un_halted_prob * (1 - lambda_n)

#

Halt based on halting probability $λ_{n}$

159            halt = torch.bernoulli(lambda_n) * (1 - halted)

#

Collect $p_{n}$ and $\overset{y}{^}_{n}$

162            p.append(p_n)
163            y.append(y_n)

#

Update $p_{m}$ and $\overset{y}{^}_{m}$ based on what was halted at current step $n$

166            p_m = p_m * (1 - halt) + p_n * halt
167            y_m = y_m * (1 - halt) + y_n * halt

#

Update halted samples

170            halted = halted + halt

#

Get next state $h_{n + 1} = s_{h} (x, h_{n})$

172            h = self.gru(x, h)

#

Stop the computation if all samples have halted

175            if self.is_halt and halted.sum() == batch_size:
176                break

#

179        return torch.stack(p), torch.stack(y), p_m, y_m

#

Reconstruction loss

$L_{R ec} = n = 1 \sum N p_{n} L (y, \overset{y}{^}_{n})$

$L$ is the normal loss function between target $y$ and prediction $\overset{y}{^}_{n}$ .

182class ReconstructionLoss(Module):

#

loss_func is the loss function $L$

191    def __init__(self, loss_func: nn.Module):

#

195        super().__init__()
196        self.loss_func = loss_func

#

p is $p_{1} \dots p_{N}$ in a tensor of shape [N, batch_size]
y_hat is $\overset{y}{^}_{1} \dots \overset{y}{^}_{N}$ in a tensor of shape [N, batch_size, ...]
y is the target of shape [batch_size, ...]

198    def forward(self, p: torch.Tensor, y_hat: torch.Tensor, y: torch.Tensor):

#

The total $\sum_{n = 1}^{N} p_{n} L (y, \overset{y}{^}_{n})$

206        total_loss = p.new_tensor(0.)

#

Iterate upto $N$

208        for n in range(p.shape[0]):

#

$p_{n} L (y, \overset{y}{^}_{n})$ for each sample and the mean of them

210            loss = (p[n] * self.loss_func(y_hat[n], y)).mean()

#

Add to total loss

212            total_loss = total_loss + loss

#

215        return total_loss

#

Regularization loss

$L_{R e g} = K L (p_{n} ∥ p_{G} (λ_{p}))$

$K L$ is the Kullback–Leibler divergence.

$p_{G}$ is the Geometric distribution parameterized by $λ_{p}$ . $λ_{p}$ has nothing to do with $λ_{n}$ ; we are just sticking to same notation as the paper. $P r_{p_{G} (λ_{p})} (X = k) = (1 - λ_{p})^{k} λ_{p}$ .

The regularization loss biases the network towards taking $\frac{1}{λ _{p}}$ steps and incentivies non-zero probabilities for all steps; i.e. promotes exploration.

218class RegularizationLoss(Module):

#

lambda_p is $λ_{p}$ - the success probability of geometric distribution
max_steps is the highest $N$ ; we use this to pre-compute $p_{G} (λ_{p})$

234    def __init__(self, lambda_p: float, max_steps: int = 1_000):

#

239        super().__init__()

#

Empty vector to calculate $p_{G} (λ_{p})$

242        p_g = torch.zeros((max_steps,))

#

$(1 - λ_{p})^{k}$

244        not_halted = 1.

#

Iterate upto max_steps

246        for k in range(max_steps):

#

$P r_{p_{G} (λ_{p})} (X = k) = (1 - λ_{p})^{k} λ_{p}$

248            p_g[k] = not_halted * lambda_p

#

Update $(1 - λ_{p})^{k}$

250            not_halted = not_halted * (1 - lambda_p)

#

Save $P r_{p_{G} (λ_{p})}$

253        self.p_g = nn.Parameter(p_g, requires_grad=False)

#

KL-divergence loss

256        self.kl_div = nn.KLDivLoss(reduction='batchmean')

#

p is $p_{1} \dots p_{N}$ in a tensor of shape [N, batch_size]

258    def forward(self, p: torch.Tensor):

#

Transpose p to [batch_size, N]

263        p = p.transpose(0, 1)

#

Get $P r_{p_{G} (λ_{p})}$ upto $N$ and expand it across the batch dimension

265        p_g = self.p_g[None, :p.shape[1]].expand_as(p)

#

Calculate the KL-divergence. The PyTorch KL-divergence implementation accepts log probabilities.

270        return self.kl_div(p.log(), p_g)