This is a sample blog post demonstrating the Jekyll setup with mathematical notation support. You can write technical content with inline math like $E = mc^2$ or display equations.

The core idea behind differentiable neural architecture search (DARTS) can be expressed through a bilevel optimization problem:

\[\min_{\alpha} \mathcal{L}_{val}(w^*(\alpha), \alpha)\]

subject to:

\[w^*(\alpha) = \arg\min_{w} \mathcal{L}_{train}(w, \alpha)\]

where $\alpha$ represents the architecture parameters and $w$ represents the network weights.

Search Space Relaxation

Instead of searching over discrete architectures, we relax the search space to be continuous. For a given edge $(i,j)$, we compute:

\[\bar{o}^{(i,j)} = \sum_{o \in \mathcal{O}} \frac{\exp(\alpha_o^{(i,j)})}{\sum_{o' \in \mathcal{O}} \exp(\alpha_{o'}^{(i,j)})} o(x^{(i)})\]

This allows gradient-based optimization of the architecture parameters.

Code Implementation

Here’s a simplified implementation in PyTorch:

class DifferentiableOp(nn.Module):
    def __init__(self, operations):
        super().__init__()
        self.ops = nn.ModuleList(operations)
        self.alpha = nn.Parameter(torch.randn(len(operations)))

    def forward(self, x):
        weights = F.softmax(self.alpha, dim=0)
        return sum(w * op(x) for w, op in zip(weights, self.ops))

Convergence Analysis

The convergence rate of the alternating optimization can be bounded by:

\[\|\alpha_t - \alpha^*\| \leq \rho^t \|\alpha_0 - \alpha^*\|\]

where $\rho < 1$ depends on the smoothness and strong convexity properties of the loss landscape.

This framework enables efficient exploration of vast architecture spaces while maintaining theoretical guarantees on convergence.