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quantized-evolution-strategies Fine-tune quantized LLMs directly in low-precision discrete parameter space using evolution strategies with accumulated error feedback. Overcome gradient stagnation in quantized models by accumulating fractional updates using Delta-Sigma modulation, achieving significant improvements in INT4 quantized models without full-precision gradients.
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name quantized-evolution-strategies title Quantized Evolution Strategies: High-precision Fine-tuning of Quantized LLMs version 0.0.2 engine skillxiv-v0.0.2-claude-opus-4.6 license MIT url https://arxiv.org/abs/2602.03120 keywords ["Quantization","Fine-Tuning","Evolution Strategies","Gradient Stagnation","Memory Efficiency"] description Fine-tune quantized LLMs directly in low-precision discrete parameter space using evolution strategies with accumulated error feedback. Overcome gradient stagnation in quantized models by accumulating fractional updates using Delta-Sigma modulation, achieving significant improvements in INT4 quantized models without full-precision gradients.
Quantized Evolution Strategies: High-precision Fine-tuning of Quantized LLMs
Problem Context
Quantized language models (INT4, INT8) reduce memory but create a fundamental optimization problem: gradient updates often fall below quantization thresholds, causing training to stagnate. Standard backprop doesn't work because discrete rounding eliminates infinitesimal signals. Quantized Evolution Strategies (QES) solves this using population-based perturbations and accumulated error feedback inspired by signal processing.
Core Concept
QES operates in three phases: (1) estimate gradients using population-based perturbations without backprop, (2) accumulate error residuals at each iteration (Delta-Sigma modulation), (3) make discrete updates only when accumulated error crosses rounding threshold. This transforms infinitesimal signals into discrete changes within quantized parameter space.
Architecture Overview
Population-based gradients : Estimate gradients via finite differences on random perturbationsError accumulation : Track fractional updates that don't cross quantization boundariesDelta-Sigma feedback : Accumulated residuals trigger discrete parameter changesStateless replay : Reconstruct optimization history from seeds, not full stateDirect quantization optimization : No full-precision shadow parameters
Implementation
Step 1: Population-based gradient estimation
import torch
import numpy as np
from typing import Tuple
class PopulationGradientEstimator :
"""Estimate gradients from function evaluations (no backprop)."""
def __init__ (
self,
pop_size: int = 32 ,
sigma: float = 0.01 ,
learning_rate: float = 1e-3
):
self .pop_size = pop_size
self .sigma = sigma
self
def
estimate_gradients
self,
model: torch.nn.Module,
reward_fn,
batch: dict
dict
"""
Estimate gradients using population sampling.
Args:
model: Quantized language model
reward_fn: Function returning scalar reward
batch: Input batch
Returns:
estimated_grads: Per-parameter gradient estimates
"""
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Step 2: Accumulated error feedback (Delta-Sigma modulation) class AccumulatedErrorFeedback :
"""Accumulate fractional updates until crossing quantization threshold."""
def __init__ (self, quantization_bits: int = 4 ):
self .quantization_bits = quantization_bits
self .quantization_step = 1.0 / (2 ** quantization_bits)
self .error_buffer = {}
def apply_accumulated_update (
self,
param_tensor: torch.Tensor,
param_name: str ,
gradient: torch.Tensor,
learning_rate: float
) -> Tuple [torch.Tensor, float ]:
"""
Apply update with error accumulation.
Strategy:
- Compute update: delta = lr * gradient
- Add to error buffer: error[name] += delta
- When |error| > quantization_step: make discrete update
"""
if param_name not in self .error_buffer:
self .error_buffer[param_name] = torch.zeros_like(param_tensor)
proposed_update = learning_rate * gradient
self .error_buffer[param_name].add_(proposed_update)
error = self .error_buffer[param_name]
discrete_update = torch.sign(error) * self .quantization_step
crosses_threshold = torch.abs (error) >= self .quantization_step
actual_update = discrete_update.clone()
actual_update[~crosses_threshold] = 0
param_tensor.add_(actual_update)
self .error_buffer[param_name][crosses_threshold] = 0
num_updates = crosses_threshold.sum ().item()
return param_tensor, num_updates
Step 3: Stateless seed replay optimization class StatelessSeedReplay :
"""Reconstruct optimization state from random seeds without memory overhead."""
def __init__ (self ):
self .seed_history = []
self .reward_history = []
def save_iteration_state (self, seed: int , reward: float ):
"""Save only seed and reward, not full gradient state."""
self .seed_history.append(seed)
self .reward_history.append(reward)
def reconstruct_gradients (
self,
model: torch.nn.Module,
iteration: int ,
sigma: float = 0.01
) -> dict :
"""
Reconstruct gradients at specific iteration from saved seeds.
Avoids storing full gradient history.
"""
torch.manual_seed(self .seed_history[iteration])
params = {name: p for name, p in model.named_parameters()}
estimated_grads = {}
baseline_reward = self .reward_history[iteration]
for i in range (32 ):
pert_dict = {}
for name in params.keys():
pert = sigma * torch.randn_like(params[name])
pert_dict[name] = pert
if i < len (self .reward_history):
reward = self .reward_history[i + iteration * 32 ]
else :
reward = baseline_reward
for name, pert in pert_dict.items():
advantage = reward - baseline_reward
if name not in estimated_grads:
estimated_grads[name] = torch.zeros_like(params[name])
estimated_grads[name].add_(advantage * pert)
for name in estimated_grads:
estimated_grads[name].div_(32 * sigma ** 2 )
return estimated_grads
Step 4: QES training step class QuantizedEvolutionStrategies :
"""Full QES optimizer for quantized model fine-tuning."""
def __init__ (
self,
model: torch.nn.Module,
reward_fn,
quantization_bits: int = 4 ,
pop_size: int = 32 ,
learning_rate: float = 1e-3 ,
use_seed_replay: bool = True
):
self .model = model
self .reward_fn = reward_fn
self .pop_size = pop_size
self .learning_rate = learning_rate
self .grad_estimator = PopulationGradientEstimator(
pop_size=pop_size, sigma=0.01 , learning_rate=learning_rate
)
self .error_feedback = AccumulatedErrorFeedback(quantization_bits)
self .seed_replay = StatelessSeedReplay() if use_seed_replay else None
def training_step (self, batch: dict ) -> dict :
"""Single QES training step."""
estimated_grads = self .grad_estimator.estimate_gradients(
self .model, self .reward_fn, batch
)
total_updates = 0
for name, param in self .model.named_parameters():
if name in estimated_grads:
grad = estimated_grads[name]
_, num_updates = self .error_feedback.apply_accumulated_update(
param.data, name, grad, self .learning_rate
)
total_updates += num_updates
reward = self .reward_fn(self .model, batch)
return {
'reward' : reward,
'num_updates' : total_updates,
'update_ratio' : total_updates / sum (p.numel() for p in self .model.parameters())
}
Step 5: Fine-tuning loop def fine_tune_quantized_llm (
model: torch.nn.Module,
train_loader,
verifier,
num_steps: int = 1000 ,
quantization_bits: int = 4 ,
learning_rate: float = 1e-3 ,
device: str = 'cuda'
):
"""
Fine-tune quantized LLM using QES.
Args:
model: Quantized language model (INT4 or INT8)
train_loader: Training data
verifier: Reward function (task accuracy or loss)
num_steps: Optimization steps
quantization_bits: Bit-width of quantization
learning_rate: ES learning rate
"""
def reward_fn (model, batch ):
"""Compute reward on batch."""
with torch.no_grad():
outputs = model(**batch)
loss = outputs.loss if hasattr (outputs, 'loss' ) else outputs[0 ]
return -loss.item()
qes = QuantizedEvolutionStrategies(
model, reward_fn,
quantization_bits=quantization_bits,
pop_size=32 ,
learning_rate=learning_rate
)
for step in range (num_steps):
batch = next (iter (train_loader))
batch = {k: v.to(device) for k, v in batch.items()}
metrics = qes.training_step(batch)
if (step + 1 ) % 100 == 0 :
print (f"Step {step + 1 } : "
f"Reward={metrics['reward' ]:.4 f} , "
f"Updates={metrics['num_updates' ]} , "
f"UpdateRatio={metrics['update_ratio' ]:.4 f} " )
return model
Practical Guidance When to use : Fine-tuning quantized models (INT4, INT8) where gradient-based methods fail due to discrete rounding
pop_size : 16-64 (larger = better gradient estimates but slower)sigma : 0.005-0.02 (perturbation scale)learning_rate : 1e-4 to 1e-2 (depends on pop_size)quantization_bits : 4 (INT4) or 8 (INT8)
Overcomes gradient stagnation in quantized models
No full-precision shadow parameters (memory efficient)
Population-based approach robust to noise
Error accumulation ensures progress despite discrete constraints
pop_size too small → noisy gradients
learning_rate too high → thrashing in discrete space
Not validating that updates actually improve loss
Using on non-quantized models (wastes computation)
Scaling : Linear in pop_size and model size. Efficient for INT4 models.
Reference Paper: https://arxiv.org/abs/2602.03120
Related work: Evolution strategies, quantization, fine-tuning, Delta-Sigma modulation
Benchmarks: Arithmetic reasoning on INT4 quantized models
