Understand why math reasoning improvements don't always transfer to general capabilities. Use RL-based training instead of SFT to preserve representation structure and enable broader generalization.
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Understand why math reasoning improvements don't always transfer to general capabilities. Use RL-based training instead of SFT to preserve representation structure and enable broader generalization.
Math Reasoning Transfer: When Specialization Preserves Generality
Training a language model to excel at mathematics—adding chain-of-thought prompting, reinforcement learning on mathematical reasoning, instruction tuning on problem sets—seems like an obvious way to improve general reasoning. Yet most models trained this way experience "catastrophic forgetting": they become better at math but worse at reading comprehension, commonsense reasoning, and other non-mathematical tasks. This creates a dilemma: improve specialized math performance at the cost of general capabilities, or maintain general abilities while sacrificing math performance.
The core problem is not math training itself, but how the model is trained. Supervised fine-tuning (SFT) on math examples causes the model's internal representations to drift away from their original structure—the patterns that made the model generally useful in the first place. Reinforcement learning (RL), by contrast, adjusts the model's behavior while preserving its representation structure through KL regularization, allowing math improvements to coexist with general capabilities.
Core Concept
The key insight is that representation structure matters more than task data. When you SFT on math data:
The model's hidden layers shift to prioritize mathematical patterns
Non-mathematical reasoning patterns (needed for other tasks) degrade
Transfer is blocked because the general reasoning capability is damaged
When you RL-train on math with KL regularization:
The model learns to prefer better math outputs
But KL keeps hidden representations close to the original
General reasoning patterns remain intact, enabling transfer
The surprising finding: RL-trained models generalize better than SFT-trained models regardless of model size. A 3B RL-trained model transfers better than a 14B SFT-trained model on the same math reasoning task.
Architecture Overview
The approach uses diagnostic analysis to understand representation drift:
On-Policy RL Training: Use GRPO, PPO, or DPO with KL regularization to optimize math reasoning while constraining representation drift
Representation Analysis: Track how model hidden states evolve during training using PCA and dimensionality reduction
Token Distribution Analysis: Monitor token probability distributions (entropy, rank changes) to detect catastrophic forgetting
Step 1: Measure baseline transferability before training
Establish baseline performance on both math and general tasks, computing a transferability index.
import numpy as np
from scipy.spatial.distance import pdist, squareform
defevaluate_baseline_capabilities(model, tokenizer,
math_eval_data, general_eval_data):
"""
Evaluate the model on both math and general tasks before any training.
Establish baseline for comparison.
"""
math_scores = []
general_scores = []
# Evaluate on mathfor prompt, correct_answer in math_eval_data:
input_ids = tokenizer.encode(prompt, return_tensors='pt').to(device)
output_ids = model.generate(input_ids, max_length=256)
output_text = tokenizer.decode(output_ids[0])
is_correct = evaluate_mathematical_correctness(output_text, correct_answer)
math_scores.append(float(is_correct))
# Evaluate on general tasksfor prompt, expected_answer in general_eval_data:
input_ids = tokenizer.encode(prompt, return_tensors='pt').to(device)
output_ids = model.generate(input_ids, max_length=256)
output_text = tokenizer.decode(output_ids[0])
is_correct = evaluate_general_correctness(output_text, expected_answer)
general_scores.append(float(is_correct))
return {
'math_baseline': np.mean(math_scores),
'general_baseline': np.mean(general_scores)
}
defcompute_transferability_index(math_improvement, general_change):
"""
Compute a metric that normalizes transfer across benchmarks.
Positive = good transfer; negative = catastrophic forgetting.
"""# Normalize math improvement to 0-1 range (assume max possible is 50% improvement)
math_gain = min(math_improvement / 0.5, 1.0)
# Normalize general change to -1 to 0 (should be non-negative)
general_delta = min(max(general_change, -1.0), 0.0) # Capped at -1 for catastrophic failure# Transferability: combination of math gains and preserved general capabilities# RL should have positive general_delta, SFT often has negative
transferability = math_gain + max(0, general_delta)
return transferability
Step 2: Track representation drift during training
Use PCA and KL-divergence to monitor how model representations change.
deftrack_representation_drift(model, reference_model, eval_prompts, layer_idx=-2):
"""
Measure how much the model's internal representations have shifted
compared to the reference (pre-training) model.
Uses PCA dimensionality reduction to visualize drift.
"""
current_activations = []
reference_activations = []
for prompt in eval_prompts[:100]: # Use subset for efficiency
input_ids = tokenizer.encode(prompt, return_tensors='pt').to(device)
# Extract hidden states from current modelwith torch.no_grad():
outputs = model(input_ids, output_hidden_states=True)
current_hidden = outputs.hidden_states[layer_idx] # [batch, seq, hidden]
current_activations.append(current_hidden.mean(dim=1).cpu().numpy()) # Average over sequence# Extract from reference modelwith torch.no_grad():
ref_outputs = reference_model(input_ids, output_hidden_states=True)
ref_hidden = ref_outputs.hidden_states[layer_idx]
reference_activations.append(ref_hidden.mean(dim=1).cpu().numpy())
# Stack activations
current_activations = np.vstack(current_activations) # [num_prompts, hidden_dim]
reference_activations = np.vstack(reference_activations)
# Compute PCA distance: how much do the principal components differ?from sklearn.decomposition import PCA
pca_current = PCA(n_components=10)
pca_reference = PCA(n_components=10)
current_components = pca_current.fit_transform(current_activations)
reference_components = pca_reference.fit_transform(reference_activations)
# Procrustes distance: best alignment between two matrices
drift_distance = procrustes_distance(current_components, reference_components)
# Also compute KL divergence of token distributions# (does the model still generate similar token probabilities?)
kl_divergences = []
for prompt in eval_prompts[:50]:
input_ids = tokenizer.encode(prompt, return_tensors='pt').to(device)
with torch.no_grad():
current_logits = model(input_ids).logits[:, -1, :] # Last position
current_probs = torch.softmax(current_logits, dim=-1)
ref_logits = reference_model(input_ids).logits[:, -1, :]
ref_probs = torch.softmax(ref_logits, dim=-1)
# KL divergence between distributions
kl = torch.sum(ref_probs * (torch.log(ref_probs) - torch.log(current_probs)))
kl_divergences.append(kl.item())
return {
'representation_drift': drift_distance,
'kl_divergence': np.mean(kl_divergences),
'is_drifting': drift_distance > 0.3
}
Step 3: Train with RL using KL-regularized objectives
Use GRPO with KL penalty to optimize math performance while preserving general capabilities.
deftrain_math_with_rl_kl_regularization(model, tokenizer,
math_train_data, general_train_data,
math_reward_fn, learning_rate=1e-5,
kl_weight=0.1, num_steps=1000):
"""
Train on math using reinforcement learning with KL regularization.
KL weight controls how much the model can deviate from the reference.
"""
optimizer = torch.optim.AdamW(model.parameters(), lr=learning_rate)
reference_model = copy.deepcopy(model) # Frozen reference for KLfor step inrange(num_steps):
# Gather math training batch
math_batch = random.sample(math_train_data, 16)
general_batch = random.sample(general_train_data, 8)
# Process math examples for RL
math_losses = []
for prompt, correct_answer in math_batch:
input_ids = tokenizer.encode(prompt, return_tensors='pt').to(device)
# Generate multiple samples
samples = []
for _ inrange(4):
output_ids = model.generate(
input_ids,
max_length=256,
temperature=1.0, # Higher temperature for on-policy sampling
num_return_sequences=1
)
samples.append(output_ids[0])
# Score samples with reward function
rewards = [math_reward_fn(tokenizer.decode(s), correct_answer)
for s in samples]
rewards = torch.tensor(rewards, device=device)
# GRPO loss: optimize probability of high-reward samples# Log probability of each sample
log_probs = []
for sample in samples:
log_prob = compute_log_probability(model, input_ids, sample)
log_probs.append(log_prob)
log_probs = torch.stack(log_probs)
# Advantage: rewards relative to mean
advantages = rewards - rewards.mean()
grpo_loss = -(log_probs * advantages).mean()
# KL penalty: divergence from reference modelwith torch.no_grad():
ref_log_probs = [compute_log_probability(reference_model, input_ids, s)
for s in samples]
ref_log_probs = torch.stack(ref_log_probs)
kl_penalty = (log_probs - ref_log_probs).mean()
total_loss = grpo_loss + kl_weight * kl_penalty
math_losses.append(total_loss)
# Add general task examples to constrain drift
general_losses = []
for prompt, expected in general_batch:
input_ids = tokenizer.encode(prompt, return_tensors='pt').to(device)
labels = tokenizer.encode(expected, return_tensors='pt')[0].to(device)
outputs = model(input_ids, labels=labels)
general_losses.append(outputs.loss)
# Combine losses: math RL + general SFT
total_loss = sum(math_losses) / len(math_losses) + \
0.5 * sum(general_losses) / len(general_losses)
optimizer.zero_grad()
total_loss.backward()
torch.nn.utils.clip_grad_norm_(model.parameters(), 1.0)
optimizer.step()
if step % 100 == 0:
print(f"Step {step}: Math loss {sum(math_losses)/len(math_losses):.4f}, "f"General loss {sum(general_losses)/len(general_losses):.4f}")
return model
defcompute_log_probability(model, input_ids, sample_ids, device='cuda'):
"""
Compute the log probability of generating sample_ids given input_ids.
"""# Concatenate input and sample
full_ids = torch.cat([input_ids, sample_ids.unsqueeze(0)], dim=1)
with torch.no_grad():
outputs = model(full_ids, output_hidden_states=False)
logits = outputs.logits
# Get log probabilities for the sample part
sample_start = input_ids.shape[1]
sample_logits = logits[:, sample_start-1:-1, :] # Shift by 1 for next-token prediction
log_probs = torch.nn.functional.log_softmax(sample_logits, dim=-1)
# Extract log probs for the actual samples
log_prob_sum = 0.0for i, token_id inenumerate(sample_ids):
log_prob = log_probs[0, i, token_id]
log_prob_sum += log_prob.item()
return log_prob_sum / len(sample_ids)
Step 4: Measure and report transferability results
Compare RL-trained and SFT-trained models on transfer metrics.
defcompare_training_methods(model_sft, model_rl, math_eval_data,
general_eval_data):
"""
Compare how well SFT-trained vs RL-trained models transfer.
"""
results = {}
# Evaluate both modelsfor model_name, model in [('SFT', model_sft), ('RL', model_rl)]:
math_scores = []
general_scores = []
for prompt, answer in math_eval_data:
input_ids = tokenizer.encode(prompt, return_tensors='pt').to(device)
output_ids = model.generate(input_ids, max_length=256)
output = tokenizer.decode(output_ids[0])
score = float(evaluate_mathematical_correctness(output, answer))
math_scores.append(score)
for prompt, answer in general_eval_data:
input_ids = tokenizer.encode(prompt, return_tensors='pt').to(device)
output_ids = model.generate(input_ids, max_length=256)
output = tokenizer.decode(output_ids[0])
score = float(evaluate_general_correctness(output, answer))
general_scores.append(score)
results[model_name] = {
'math_accuracy': np.mean(math_scores),
'general_accuracy': np.mean(general_scores),
'transferability': compute_transferability_index(
np.mean(math_scores) - baseline_math,
np.mean(general_scores) - baseline_general
)
}
return results
Practical Guidance
Hyperparameter
Recommended Value
Notes
KL weight
0.05-0.15
Higher = preserve representations; lower = focus on task
Ratio of math to general data
0.5-0.7
50-70% math examples, rest general
RL temperature
0.8-1.2
Controls diversity during sampling
Training duration
2-5K steps
Diminishing returns after 3K
Reference model update
Never
Use frozen initial model for KL
When to use RL-based reasoning training:
You want to improve math reasoning without hurting general capabilities
You have multiple task domains and need balanced performance
Representation drift matters more than raw task performance
You're willing to spend more compute on RL training
When NOT to use RL-based training:
You only care about a single specialized task (use SFT)
Compute budget is critical (RL is 30-50% more expensive than SFT)
Your task requires very high specialization and general forgetting is acceptable
You don't have multiple evaluation domains to assess transfer
Common pitfalls:
KL weight too high: Model won't improve much. Start with 0.1 and decrease if needed.
Reference model drift: Never update the reference model; it's the anchor for KL.
Insufficient general data: If the model still forgets general tasks, increase general task ratio to 50%.
Reward signal too noisy: If RL training is unstable, use a simpler reward function or increase batch size for more stable advantage estimation.
Reference
Does Math Reasoning Improve General LLM Capabilities? Understanding Transferability of LLM Reasoning
https://arxiv.org/abs/2507.00432
