Soft Cell Assignment¶

One of the central technical challenges in FAMAIL is that the objective function depends on discrete grid cell counts (e.g., "how many pickups are in cell $(i, j)$?"), but gradient-based optimization requires continuous, differentiable functions. Assigning a pickup to a single grid cell is a step function — the gradient is zero almost everywhere.

The Problem¶

When a pickup location sits at continuous coordinates $(3.2, 7.8)$, the hard assignment says it belongs to cell $(3, 7)$. But if we shift the location slightly to $(3.6, 7.8)$, it still maps to cell $(3, 7)$ — the gradient provides no signal about how to move toward a different cell. Only at the exact boundary does the assignment change, and this produces an undefined (discontinuous) gradient.

The Solution¶

Soft cell assignment replaces the hard (one-cell) assignment with a probability distribution over nearby cells. Instead of saying "this pickup is in cell $(3, 7)$," we say "this pickup has probability 0.82 of being in cell $(3, 7)$, probability 0.08 in cell $(3, 8)$, probability 0.05 in cell $(4, 7)$," and so on.

This is implemented as a Gaussian softmax over a local neighborhood:

$$\sigma_c(\mathbf{p};\, \tau) = \frac{\exp\!\left(-\|\mathbf{p} - \mathbf{c}\|^2 \;/\; 2\tau^2\right)}{\displaystyle\sum_{c' \in \mathcal{N}} \exp\!\left(-\|\mathbf{p} - \mathbf{c}'\|^2 \;/\; 2\tau^2\right)}$$

where:

$\mathbf{p}$ is the continuous pickup location
$\mathbf{c}$ is the center of a grid cell
$\mathcal{N}$ is the 5 × 5 neighborhood around the original cell
$\tau$ is the temperature parameter

Temperature Annealing¶

The temperature $\tau$ controls how "soft" or "hard" the assignment is:

Temperature	Behavior	When Used
High ($\tau \to \infty$)	Probability spread uniformly across neighborhood	Not used (theoretical limit)
Moderate ($\tau = 1.0$)	Smooth distribution; broad gradients	Early iterations (exploration)
Low ($\tau = 0.1$)	Concentrated on nearest cell; nearly hard assignment	Late iterations (precision)

During optimization, the temperature follows an exponential decay schedule:

$$\tau_t = \tau_{\max} \cdot \left(\frac{\tau_{\min}}{\tau_{\max}}\right)^{t/T}$$

from $\tau_{\max} = 1.0$ to $\tau_{\min} = 0.1$. This creates a natural curriculum:

Early iterations: Soft assignments produce non-zero gradients for all nearby cells, allowing the optimizer to explore broadly and "see" many candidate locations.
Late iterations: Assignments sharpen to approximate the discrete reality, ensuring the final modified location corresponds to a definite grid cell.

The Full Gradient Path¶

Soft cell assignment creates a differentiable chain from pickup location to objective value:

Pickup location (continuous, differentiable)
        ↓
Soft cell probabilities (Gaussian softmax)
        ↓
Differentiable pickup counts per cell
        ↓
Spatial fairness (Gini)  +  Causal fairness (R²)  +  Fidelity (discriminator)
        ↓
Combined objective L
        ↓
Gradient ∇_p L (via backpropagation)
        ↓
Perturbation δ = clip(α · sign(∇_p L), -ε, ε)

Why this matters

Without soft cell assignment, FAMAIL could not use gradient-based optimization at all. The entire pipeline — from continuous pickup locations through discrete cell counts to fairness metrics — would be non-differentiable. This module is the bridge that makes end-to-end learning possible.

Temperature	Behavior	When Used
High (\(\tau \to \infty\))	Probability spread uniformly across neighborhood	Not used (theoretical limit)
Moderate (\(\tau = 1.0\))	Smooth distribution; broad gradients	Early iterations (exploration)
Low (\(\tau = 0.1\))	Concentrated on nearest cell; nearly hard assignment	Late iterations (precision)