I'm reading the paper Neural Ordinary Differential Equations and I have a simple question about adjoint method. When we train NODE, it uses a blackbox ODESolver to compute gradients through model parameters, hidden states, and time. It uses another quantity $\mathbf{a}(t) = \partial L / \partial \mathbf{z}(t)$ called adjoint, which also satisfies another ODE. As I understand, the authors build a single ODE that computes all the gradients $\partial L / \partial \mathbf{z}(t_{0})$ and $\partial L / \partial \theta$ by solving that single ODE. However, I can't understand how do we know the value $\partial L / \partial \mathbf{z}(t_1)$ which corresponds to the initial condition for the ODE corresponds to the adjoint. I'm using this tutorial as a reference, and it defines custom forward and backward methods for solving ODE. However, for the backward computation (especially ODEAdjoint class in the tutorial) we need to pass $\partial L / \partial \mathbf{z}$ for backpropagation, and this enables us to compute $\partial L / \partial \mathbf{z}(t_i)$ from $\partial L / \partial \mathbf{z}(t_{i+1})$, but we still need to know the adjoint value $\partial L / \partial \mathbf{z}(t_N)$. I do not understand well about how pytorch's autograd package works, and this seems to be a barrier to understand this. Could anyone explain how it operates, and where $\partial L / \partial \mathbf{z}(t_1)$ (or $\partial L / \partial \mathbf{z}(t_N)$ if this is more comfortable) comes from? Thanks in advance.

Here's my guess for the initial adjoint from simple example. Let $d\mathbf{z}/dt = Az$ be a 2-dim linear ODE with given $A \in \mathbb{R}^{2\times 2}$. If we use Euler's method as a ODE solver, then the estimate for $z(t_1)$ is explicitly given as $$\hat{\mathbf{z}}(t_1) = \mathrm{ODESolve}(\mathbf{z}(t_0), f, t_0, t_1, \theta))= \left(I + \frac{t_1 - t_0}{N}A\right)^{N} \mathbf{z}(t_0) $$ where $N$ is the number of steps for Euler's method (so that $h = (t_1 - t_0) /N$ is the step size). If we use MSE loss for training, then the loss will be $$ L(\mathbf{z}(t_1)) = \Bigl|\Bigl| \mathbf{z}_1 - \left(I + \frac{t_1 - t_0}{N}A\right)^N\mathbf{z}(t_0)\Bigr|\Bigr|_2^2 $$ where $\mathbf{z}_1$ is the true value at time $t_1$, which is $\mathbf{z}_1 = e^{A(t_1 - t_0)}\mathbf{z}(t_0)$. Since adjoint $\mathbf{a}(t) = \partial L / \partial \mathbf{z}(t)$ satisfies $$\frac{d\mathbf{a}(t)}{dt} = -\mathbf{a}(t)^{T} \frac{\partial f(\mathbf{z}(t), t, \theta)}{\partial \mathbf{z}} = \mathbf{0},$$ $\mathbf{a}(t)$ is constant and we get $\mathbf{a}(t_0) = \mathbf{a}(t_1)$. So we do not need to use augmented ODE for computing $\mathbf{a}(t)$. However, I still don't know what $\mathbf{a}(t_1) = \partial L / \partial \mathbf{z}(t_1)$ should be. If my understanding is correct, since $L = ||\mathbf{z}_1 - \mathbf{z}(t_1)||^{2}_{2}$, it seems that the answer might be $$ \frac{\partial L}{\partial \mathbf{z}(t_1)} = 2(\mathbf{z}(t_1) - \mathbf{z}_1). $$ However, this doesn't seem to be true: if it is, and if we have multiple datapoints at $t_1, t_2, \dots, t_N$, then the loss is $$ L = \frac{1}{N} \sum_{i=1}^{N}||\mathbf{z}_i -\mathbf{z}(t_i)||_{2}^{2} $$ and we may have $$ \frac{\partial L}{\partial \mathbf{z}(t_i)} = \frac{2}{N} (\mathbf{z}(t_i) - \mathbf{z}_i), $$ which means that we don't need to solve ODE associated to $\mathbf{a}(t)$.


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