multivariate_regression - q_97afcf
Dataset: mvar_f991a7 Answer: C Metric: test_mse
Prompt
You are taking the ArchitectureIQ benchmark.
Each question describes one **dataset instance** and several **choices** (each choice is one **candidate**: model, optimizer, loss, and training budget). All choices train on the same dataset instance. Each choice's training budget (`total_samples_seen`) is stated in the prompt — either shared across all choices or specified per choice in the Sample budget section. Your task is to identify which choice will achieve the **best test metric** on the held-out test set after its stated training budget — using architecture and learning-mechanics reasoning only (no training results are provided).
Answer format is specified in the final section ("Your answer"): end your reply with a short `<explanation></explanation>` paragraph followed by a single letter in `<answer></answer>` tags.
## Dataset
Multivariate regression on `[0, 1]^n`: input is a vector `x ∈ R^n`, target is a scalar `y ∈ R` from a fixed symbolic expression. Train and test splits are fixed before any candidate training.
### Synthesis (PyTorch)
```python
def target(x: torch.Tensor) -> torch.Tensor:
return torch.cos(2 * torch.pi * x[:, 6]) * torch.sin(2 * torch.pi * x[:, 0]) + torch.tanh(2 * x[:, 1]) ** 2 + x[:, 7] ** 2 - torch.cos(2 * torch.pi * x[:, 6]) + torch.tanh(2 * torch.tanh(2 * x[:, 4] ** 3)) + x[:, 0] - x[:, 2] + x[:, 3] + x[:, 5] + 6
def synthesize(
*,
train_size: int = 256,
test_size: int = 256,
point_seed: int = 1636417296,
input_dim: int = 8,
domain_low: float = 0.0,
domain_high: float = 1.0,
) -> tuple[torch.Tensor, torch.Tensor, torch.Tensor, torch.Tensor]:
gen = torch.Generator().manual_seed(point_seed)
train_x = torch.rand(train_size, input_dim, generator=gen) * (domain_high - domain_low) + domain_low
test_x = torch.rand(test_size, input_dim, generator=gen) * (domain_high - domain_low) + domain_low
train_y = target(train_x)
test_y = target(test_x)
return train_x, train_y.unsqueeze(-1), test_x, test_y.unsqueeze(-1)
```
### Data splits and training protocol
- Target expression (canonical): `cos(2*pi*x6)*sin(2*pi*x0) + tanh(2*x1)**2 + x7**2 - cos(2*pi*x6) + tanh(2*tanh(2*x4**3)) + x0 - x2 + x3 + x5 + 6`
- Input dimension: 8
- Train split size: 256 fixed `(x, y)` pairs
- Test split size: 256 fixed `(x, y)` pairs (held out)
- Input domain: [0.0, 1.0] per coordinate, uniform sampling
- Point-sampling seed: 1636417296
- Evaluation: **test MSE** on the held-out split
- Reference device: cpu
## Sample budget (same for all choices)
- training_steps: 256
- batch_size: 16
- total_samples_seen: 4096 (= training_steps × batch_size)
## Evaluation metric
- Ground-truth ranking uses **test_mse** on the held-out test split.
- Each choice is trained independently for **10** seeds (`0`..`9`), one `torch.manual_seed(seed)` per run.
- Execution device: cpu.
- The correct choice has the lowest **mean** test_mse across seeds.
## Choices
### Choice A
**Model (natural language)**
- Type: MLP
- Input dimension: 8
- Depth: 5 hidden Linear layers of width 256, between the input projection and the output head (7 nn.Linear layers in total)
- Width: 256 (all hidden layers)
- Residual connections: True
- Layer norm per layer: [False, True, False, True, False]
- Activation: relu (one activation, shared by every layer)
- Initialization: PyTorch Linear defaults
**Model code**
```python
class MLPBlock(nn.Module):
"""One hidden layer: relu(x + Linear(norm(x)))."""
def __init__(self, width: int, use_layer_norm: bool) -> None:
super().__init__()
self.norm = nn.LayerNorm(width) if use_layer_norm else nn.Identity()
self.linear = nn.Linear(width, width)
self.act = nn.ReLU()
def forward(self, x: torch.Tensor) -> torch.Tensor:
h = self.linear(self.norm(x))
h = h + x
return self.act(h)
class Model(nn.Module):
def __init__(self) -> None:
super().__init__()
self.net = nn.Sequential(
nn.Linear(8, 256),
nn.ReLU(),
MLPBlock(width=256, use_layer_norm=False),
MLPBlock(width=256, use_layer_norm=True),
MLPBlock(width=256, use_layer_norm=False),
MLPBlock(width=256, use_layer_norm=True),
MLPBlock(width=256, use_layer_norm=False),
nn.Linear(256, 1),
)
def forward(self, x: torch.Tensor) -> torch.Tensor:
return self.net(x)
```
**Optimizer**
- Optimizer: Adam
- Learning rate: 0.0001
- Weight decay: 1e-05
- Betas: [0.9, 0.999]
```python
def build_optimizer(model: nn.Module) -> torch.optim.Optimizer:
params = model.parameters()
return torch.optim.Adam(params, lr=0.0001, betas=(0.9, 0.999), weight_decay=1e-05)
```
**Loss**
- Loss: mean squared error (MSE) on the minibatch
```python
def loss_fn(model: nn.Module, pred: torch.Tensor, target: torch.Tensor) -> torch.Tensor:
return torch.mean((pred - target) ** 2)
```
### Choice B
**Model (natural language)**
- Type: MLP
- Input dimension: 8
- Depth: 4 hidden Linear layers of width 256, between the input projection and the output head (6 nn.Linear layers in total)
- Width: 256 (all hidden layers)
- Residual connections: True
- Layer norm per layer: [False, False, False, True]
- Activation: silu (one activation, shared by every layer)
- Initialization: PyTorch Linear defaults
**Model code**
```python
class MLPBlock(nn.Module):
"""One hidden layer: silu(x + Linear(norm(x)))."""
def __init__(self, width: int, use_layer_norm: bool) -> None:
super().__init__()
self.norm = nn.LayerNorm(width) if use_layer_norm else nn.Identity()
self.linear = nn.Linear(width, width)
self.act = nn.SiLU()
def forward(self, x: torch.Tensor) -> torch.Tensor:
h = self.linear(self.norm(x))
h = h + x
return self.act(h)
class Model(nn.Module):
def __init__(self) -> None:
super().__init__()
self.net = nn.Sequential(
nn.Linear(8, 256),
nn.SiLU(),
MLPBlock(width=256, use_layer_norm=False),
MLPBlock(width=256, use_layer_norm=False),
MLPBlock(width=256, use_layer_norm=False),
MLPBlock(width=256, use_layer_norm=True),
nn.Linear(256, 1),
)
def forward(self, x: torch.Tensor) -> torch.Tensor:
return self.net(x)
```
**Optimizer**
- Optimizer: Adam
- Learning rate: 0.0001
- Weight decay: 1e-05
- Betas: [0.9, 0.999]
```python
def build_optimizer(model: nn.Module) -> torch.optim.Optimizer:
params = model.parameters()
return torch.optim.Adam(params, lr=0.0001, betas=(0.9, 0.999), weight_decay=1e-05)
```
**Loss**
- Loss: mean squared error (MSE) on the minibatch
```python
def loss_fn(model: nn.Module, pred: torch.Tensor, target: torch.Tensor) -> torch.Tensor:
return torch.mean((pred - target) ** 2)
```
### Choice C
**Model (natural language)**
- Type: MLP
- Input dimension: 8
- Depth: 4 hidden Linear layers of width 256, between the input projection and the output head (6 nn.Linear layers in total)
- Width: 256 (all hidden layers)
- Residual connections: False
- Layer norm per layer: [True, False, True, False]
- Activation: silu (one activation, shared by every layer)
- Initialization: PyTorch Linear defaults
**Model code**
```python
class MLPBlock(nn.Module):
"""One hidden layer: silu(Linear(norm(x)))."""
def __init__(self, width: int, use_layer_norm: bool) -> None:
super().__init__()
self.norm = nn.LayerNorm(width) if use_layer_norm else nn.Identity()
self.linear = nn.Linear(width, width)
self.act = nn.SiLU()
def forward(self, x: torch.Tensor) -> torch.Tensor:
h = self.linear(self.norm(x))
return self.act(h)
class Model(nn.Module):
def __init__(self) -> None:
super().__init__()
self.net = nn.Sequential(
nn.Linear(8, 256),
nn.SiLU(),
MLPBlock(width=256, use_layer_norm=True),
MLPBlock(width=256, use_layer_norm=False),
MLPBlock(width=256, use_layer_norm=True),
MLPBlock(width=256, use_layer_norm=False),
nn.Linear(256, 1),
)
def forward(self, x: torch.Tensor) -> torch.Tensor:
return self.net(x)
```
**Optimizer**
- Optimizer: Adam
- Learning rate: 0.0001
- Weight decay: 1e-05
- Betas: [0.9, 0.999]
```python
def build_optimizer(model: nn.Module) -> torch.optim.Optimizer:
params = model.parameters()
return torch.optim.Adam(params, lr=0.0001, betas=(0.9, 0.999), weight_decay=1e-05)
```
**Loss**
- Loss: mean squared error (MSE) on the minibatch
```python
def loss_fn(model: nn.Module, pred: torch.Tensor, target: torch.Tensor) -> torch.Tensor:
return torch.mean((pred - target) ** 2)
```
## Your answer
Choose exactly one of A, B, C. End your reply with exactly two tagged fields, in this order and with nothing after them:
<explanation>The mechanism that decides the winner, and why each other choice loses.</explanation>
<answer>the letter of your chosen option</answer>
Rules: `<answer>` must contain exactly one letter from {A, B, C}. The grader reads only the **final** `<answer>` tag in your reply.A - c_45dbe0
{
"model": {
"type": "mlp",
"depth": 5,
"width": 256,
"residual": true,
"activation": "relu",
"layer_norm": [
false,
true,
false,
true,
false
],
"input_dim": 8,
"output_dim": 1
},
"optimizer": {
"type": "Adam",
"lr": 0.0001,
"weight_decay": 1e-05,
"betas": [
0.9,
0.999
]
},
"budget": {
"training_steps": 256,
"batch_size": 16,
"total_samples_seen": 4096
}
}B - c_558774
{
"model": {
"type": "mlp",
"depth": 4,
"width": 256,
"residual": true,
"activation": "silu",
"layer_norm": [
false,
false,
false,
true
],
"input_dim": 8,
"output_dim": 1
},
"optimizer": {
"type": "Adam",
"lr": 0.0001,
"weight_decay": 1e-05,
"betas": [
0.9,
0.999
]
},
"budget": {
"training_steps": 256,
"batch_size": 16,
"total_samples_seen": 4096
}
}C - c_49113e
{
"model": {
"type": "mlp",
"depth": 4,
"width": 256,
"residual": false,
"activation": "silu",
"layer_norm": [
true,
false,
true,
false
],
"input_dim": 8,
"output_dim": 1
},
"optimizer": {
"type": "Adam",
"lr": 0.0001,
"weight_decay": 1e-05,
"betas": [
0.9,
0.999
]
},
"budget": {
"training_steps": 256,
"batch_size": 16,
"total_samples_seen": 4096
}
}
