Stability analysis and LSTMs


  • Will this necessarily be「Bounded Input Bounded Output」?
    • Guaranteed if output and hidden activations are bounded
    • But will it saturate

#Analyzing Recursion

  • Sufficient to analyze the behavior of the hidden layer since it carries the relevant information

  • Assumed linear systems

    • $$ z_{k}=W_{h} h_{k-1}+W_{x} x_{k}, \quad h_{k}=z_{k} $$
  • Sufficient to analyze the response to a single input at $t =0$ (else is zero input)

#Simple scalar linear recursion

  • $h(t) = wh(t-1) + cx(t)$
  • $h_0(t) = w^tcx(0)$
  • If $w > 1$ it will blow up

#Simple Vector linear recursion

  • $h(t) = Wh(t-1) + Cx(t)$
  • $h_0(t) = W^tCx(0)$
  • For any input, for large the length of the hidden vector will expand or contract according to the $t-$ th power of the largest eigen value of the hidden-layer weight matrix
  • If $|\lambda_{max} > 1|$ it will blow up, otherwise it will contract and shrink to 0 rapidly


  • Sigmoid: Saturates in a limited number of steps, regardless of $w$
    • To a value dependent only on $w$ (and bias, if any)
    • Rate of saturation depends on $w$
  • Tanh: Sensitive to $w$, but eventually saturates
    • “Prefers” weights close to 1.0
  • Relu: Sensitive to $w$, can blow up


  • Recurrent networks retain information from the infinite past in principle
  • In practice, they tend to blow up or forget
    • If the largest Eigen value of the recurrent weights matrix is greater than 1, the network response may blow up
    • If it’s less than one, the response dies down very quickly
  • The “memory” of the network also depends on the parameters (and activation) of the hidden units
    • Sigmoid activations saturate and the network becomes unable to retain new information
    • RELU activations blow up or vanish rapidly
    • Tanh activations are the most effective at storing memory
      • And still has very short “memory”
      • Still sensitive to Eigenvalues of $W$

#Vanishing gradient

  • A particular problem with training deep networks is the gradient of the error with respect to weights is unstable
  • For
    • $\operatorname{Div}(X)=D\left(f_{N}\left(W_{N-1} f_{N-1}\left(W_{N-2} f_{N-2}\left(\ldots W_{0} X\right)\right)\right)\right)$
  • We get
    • $\nabla_{f_{k}} \operatorname{Div}=\nabla D . \nabla f_{N} . W_{N-1} . \nabla f_{N-1} . W_{N-2} \ldots \nabla f_{k+1} W_{k}$
  • Where
    • $\nabla{f_{n}}$ is *jacobian* of $f_N()$ to its current input

#For activation

  • For RNN
    • $\nabla f_{t}\left(z_{i}\right)=\left[\begin{array}{cccc}f_{t, 1}^{\prime}\left(z_{1}\right) & 0 & \cdots & 0 \\ 0 & f_{t, 2}^{\prime}\left(z_{2}\right) & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & f_{t, N}^{\prime}\left(z_{N}\right)\end{array}\right]$
    • For vector activations: A full matrix
    • For scalar activations: A matrix where the diagonal entries are the derivatives of the activation of the recurrent hidden layer
  • The derivative (or subgradient) of the activation function is always bounded
  • Most common activation functions, such as sigmoid, tanh() and RELU have derivatives that are always less than 1
    • Multiplication by the Jacobian is always a shrinking operation
    • After a few layers the derivative of the divergence at any time is totally “forgotten”

#For weights

  • In a single-layer RNN, the weight matrices are identical
    • The conclusion below holds for any deep network, though
  • The chain product for $\nabla_{f_k} Div$ will
    • Expand $\nabla D$ along directions in which the singular values of the weight matrices are greater than 1
    • Shrink $\nabla D$ in directions where the singular values are less than 1
    • Repeated multiplication by the weights matrix will result in Exploding or vanishing gradients



  • Recurrent nets are very deep nets
  • Stuff gets forgotten in the forward pass too
    • Each weights matrix and activation can shrink components of the input
  • Need the long-term dependency
  • The memory retention of the network depends on the behavior of the weights and jacobian
  • Which in turn depends on the parameters $W$ rather than what it is trying to remember
  • We need
    • Not be directly dependent on vagaries of network parameters, but rather on input-based determination of whether it must be remembered
    • Retain memories until a switch based on the input flags them as ok to forget
      • 「Curly brace must remember until curly brace is closed」
  • LSTM
    • Address the problem of input-dependent memory behavior


  • The $\sigma$ are multiplicative gates that decide if something is important or not

#Key component

Remembered cell state

  • Mutiply is a switch
    • Should I continue remember or not? (scale up / down)
  • Acddition
    • Should I agument the memory?
  • $C_t$ is the linear history carried by the constant-error carousel
  • Carries information through, only affected by a gate
    • And addition of history, which too is gated..


  • Gates are simple sigmoidal units with outputs in the range (0,1)
  • Controls how much of the information is to be let through

#Forget gate

  • The first gate determines whether to carry over the history or to forget it
    • More precisely, how much of the history to carry over
    • Also called the “forget” gate
    • Note, we’re actually distinguishing between the cell memory $C$ and the state $h$ that is coming over time! They’re related though
      • Hidden state is compute from memory (which is stored)

#Input gate

  • The second input has two parts
    • A perceptron layer that determines if there’s something new and interesting in the input
      • 「See a curly brace」
    • A gate that decides if its worth remembering
      • 「Curly brace is in comment section, ignore it」

#Memory cell update

  • If something new and worth remembering
    • Added to the current memory cell

#Output and Output gate

  • The output of the cell
    • Simply compress it with tanh to make it lie between 1 and -1
      • Note that this compression no longer affects our ability to carry memory forward
    • Controlled by an output gate
      • To decide if the memory contents are worth reporting at this time

#The “Peephole” Connection

  • The raw memory is informative by itself and can also be input
    • Note, we’re using both $C$ and $h$



$$ \begin{array}{l} \nabla_{C_{t}} D i v=&\nabla_{h_{t}} D i v \circ\left(o_{t} \circ \tanh ^{\prime}(.)+\tanh (.) \circ \sigma^{\prime}(.) W_{C o}\right)+ \\\\ &\nabla_{C_{t+1}} D i v \circ\left(f_{t+1}+C_{t} \circ \sigma^{\prime}(.) W_{C f}+\tilde{C}_{t+1} \circ \sigma^{\prime}(.) W_{C i} \circ \tanh (.) \ldots\right) \end{array} $$ $$ \begin{aligned} \nabla_{h_{t}} D i v=& \nabla_{z_{t}} D i v \nabla_{h_{t}} z_{t}+\nabla_{C_{t+1}} D i v \circ\left(C_{t} \circ \sigma^{\prime}(.) W_{h f}+\tilde{C}_{t+1} \circ \sigma^{\prime}(.) W_{h i}\right)+\\\\ &\nabla_{C_{t+1}} D i v \circ o_{t+1} \circ \tanh ^{\prime}(.) W_{h i}+\nabla_{h_{t+1}} D i v \circ \tanh (.) \circ \sigma^{\prime}(.) W_{h o} \end{aligned} $$ And weights?

#Gated Recurrent Units

  • Combine forget and input gates
    • In new input is to be remembered, then this means old memory is to be forgotten
    • No need to compute twice
  • Don’t bother to separately maintain compressed and regular memories
    • Redundant representation


  • LSTMs are an alternative formalism where memory is made more directly dependent on the input, rather than network parameters/structure
  • Through a “Constant Error Carousel” memory structure with no weights or activations, but instead direct switching and “increment/decrement” from pattern recognizers
  • Do not suffer from a vanishing gradient problem but do suffer from exploding gradient issue


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