Day 33: Diagnostic Failure Case Analysis & Independent Mastery Project Rubric

Reference Solution:
1. MRE: $a = \text{Value}(3.0); b = \text{Value}(4.0); f = (a + b) * a; f.\text{backward}()$.
2. Because in linear graphs without branching, each node receives only one incoming gradient contribution, so = happens to produce the correct value by accident.
3. Replace self.grad = ... with self.grad += ... across all backward closures.

Reference Solution:
1. Leaf gradients will be exactly doubled ($2 \times$).
2. Default accumulation allows gradient accumulation across multiple mini-batches before calling optimizer.step(), effectively simulating larger batch sizes.
3. For vector output $\mathbf{y} \in \mathbb{R}^m$, the backward pass computes $\mathbf{v}^T J$ where $\mathbf{v} = \text{grad\_outputs}$ and $J_{ij} = \frac{\partial y_i}{\partial x_j}$.

Reference Solution:
1. Pure-Python scalar engines lack vectorization (SIMD) and GPU kernel acceleration, and create massive object overhead.
2. A contiguous tensor stores floating-point numbers in linear memory buffers manipulated by compiled C++/CUDA kernels with zero per-element Python overhead.
3. micrograd is licensed under MIT; any redistributions or derivative implementations must include the original copyright notice and permission notice.

Reference Mastery Rubric Criteria:
- Criterion 1: Complete analytical derivation of $\sum \frac{\partial L}{\partial y_j} \frac{\partial y_j}{\partial x}$ for branching nodes.
- Criterion 2: Working scalar autograd engine passing unit tests without third-party autodiff packages.
- Criterion 3: Working finite-difference check with $\text{rtol} \le 10^{-3}$ identifying intentional defects.