带递归关系的单目标优化问题中未知参数Θ的求解求助

Hey there, I’ve been totally stuck on this undetermined parameter problem for a while and I can’t even figure out where to start—really hoping someone can toss me some hints or pointers!

Here’s the problem setup:
I have a sample data point $(p_0,q_0,p_M,q_M)$, and there’s a recursive relation that connects $(p_0,q_0)$ to $(p_M,q_M)$ over M steps:

$$
\begin{cases}
p(i) = (1-a\lambda(i))p(i-1)+a\lambda(i)q(i-1) \
q(i) = b\lambda(i)p(i-1)+(1-b\lambda(i))q(i-1) \
\lambda(i) = f(p(i-1), q(i-1), \Theta)
\end{cases}
$$

With the constraints:
$$
\begin{cases}
p(0)=p_0, q(0)=q_0 \
p(M)\approx p_M, q(M)\approx q_M
\end{cases}
$$

Where $a,b$ are known non-zero constants, and $\Theta$ is the unknown parameter set I need to solve for. The function $f$ defines $\lambda(i)$ based on the previous step's $p,q$ values and the parameters $\Theta$.

Some initial thoughts I’ve tossed around (but need help refining): #

• First off, this feels like a least-squares optimization problem. I could define a loss function like $L(\Theta) = (p(M;\Theta)-p_M)^2 + (q(M;\Theta)-q_M)^2$, then find the $\Theta$ that minimizes this loss.
• Since calculating $p(M)$ and $q(M)$ requires iterating through all M steps each time, this is a black-box optimization scenario if $f$ doesn’t have a nice analytical form. For cases like that, gradient-based methods (like Adam or L-BFGS) could work if I can compute gradients, or gradient-free methods (like the Nelder-Mead simplex algorithm) if gradients are hard to derive.
• If $f$ is differentiable, I could use automatic differentiation tools (like PyTorch or TensorFlow) to compute the gradient of $L(\Theta)$ with respect to $\Theta$—that would make the optimization way more efficient than guessing blindly.
• For small M, maybe I could try expanding the recursion manually to get an explicit expression for $p(M)$ and $q(M)$ in terms of $\Theta$, but that’s obviously not feasible if M is large.
• A good initial guess for $\Theta$ (if I have any prior knowledge about its possible values) would probably help the optimization algorithm converge faster.

Does anyone have more specific tips, or know if there’s a standard approach for problems like this? I’d really appreciate any help!

备注：内容来源于stack exchange，提问作者Maxchen