Low-rank approximation techniques for optimizer states can reduce the memory footprint of agentic reinforcement learning systems used for CUDA kernel generation, enabling larger-scale exploration.

Applying low-rank approximation to optimizer states (specifically the first and second moment tensors in Adam-family optimizers) during reinforcement learning training of agents that generate CUDA kernels will reduce peak GPU memory consumption by ≥30% relative to full-rank Adam baselines, without degrading final policy reward (kernel throughput/correctness) by more than 5%, thereby enabling batch sizes or model scales that are otherwise infeasible on fixed hardware budgets. Formally: ∃ rank r < d such that using rank-r approximations of optimizer moment matrices yields memory(r) ≤ 0.70 × memory(full) and reward(r) ≥ 0.95 × reward(full) across ≥3 distinct CUDA kernel generation tasks.

MINIMUM VIABLE TEST (MVT):

• Single GPU (A100 80 GB), 3 kernel tasks (matmul, softmax, fused attention), 2 model sizes (125M, 350M parameters), 3 rank budgets (r = 0.05d, 0.10d, 0.25d), full-rank Adam baseline.
• RL algorithm: PPO with 4 parallel rollout workers.
• Training: 20,000 steps per configuration.
• Primary metrics: peak GPU memory (nvidia-smi dmon at 100 ms intervals), final reward (mean kernel throughput TFLOPS vs. cuBLAS reference, averaged over last 1,000 steps), training wall-clock time.
• Total configurations: 2 sizes × 3 ranks × 3 tasks + 2 baselines × 3 tasks = 24 runs.

FULL VALIDATION:

• Extend to 4 model sizes (125M, 350M, 1.3B, 6.7B), 5 kernel tasks (add fused LayerNorm, custom GEMM), 3 RL algorithms (PPO, GRPO, ReMax), 2 hardware types (A100, H100).
• Ablation: projection update frequency (every 50, 100, 200, 500 steps).
• Ablation: projection method (randomized SVD, power iteration, structured random projection).
• Statistical rigor: 5 seeds per configuration, Welch's t-test with Bonferroni correction (α = 0.05/n_comparisons).

# ============================================================
# LOW-RANK RL OPTIMIZER — IMPLEMENTATION SKETCH
# Target: PPO agent for CUDA kernel generation
# ============================================================

import torch
import numpy as np
from sklearn.utils.extmath import randomized_svd
from typing import Optional, Tuple

# --- 1. LOW-RANK ADAM OPTIMIZER ---

class LowRankAdam(torch.optim.Optimizer):
    """
    Adam optimizer with low-rank approximation of moment tensors.
    Replaces M1 (d1 x d2) and M2 (d1 x d2) with rank-r factors.
    Memory: O(r*(d1+d2)) vs O(d1*d2) for full-rank.
    """
    def __init__(self, params, lr=1e-4, betas=(0.9, 0.999),
                 eps=1e-8, rank_ratio=0.10,
                 projection_freq=100):
        defaults = dict(lr=lr, betas=betas, eps=eps,
                       rank_ratio=rank_ratio,
                       projection_freq=projection_freq)
        super().__init__(params, defaults)
        self.step_count = 0

    def _get_rank(self, param: torch.Tensor) -> int:
        """Compute rank r = rank_ratio * min(d1, d2)."""
        if param.dim() < 2:
            return None  # 1D params use full-rank Adam
        min_dim = min(param.shape[-2], param.shape[-1])
        return max(1, int(self.defaults['rank_ratio'] * min_dim))

    def _init_low_rank_state(self, param: torch.Tensor,
                              state: dict, rank: int):
        """Initialize U, S, V factors for moment approximation."""
        d1, d2 = param.shape[-2], param.shape[-1]
        state['m1_U'] = torch.zeros(d1, rank, device=param.device)
        state['m1_V'] = torch.zeros(rank, d2, device=param.device)
        state['m2_U'] = torch.zeros(d1, rank, device=param.device)
        state['m2_V'] = torch.zeros(rank, d2, device=param.device)
        state['step'] = 0

    def _project_to_low_rank(self, M: torch.Tensor,
                              rank: int) -> Tuple:
        """Randomized SVD projection. O(d1*d2*rank) time."""
        M_np = M.float().cpu().numpy()
        U, S, Vt = randomized_svd(M_np, n_components=rank,
                                   n_iter=2, random_state=42)
        # Absorb S into U: U_scaled = U * sqrt(S)
        S_sqrt = np.sqrt(S)
        U_scaled = torch.tensor(U * S_sqrt[None, :],
                                dtype=torch.float32,
                                device=M.device)
        V_scaled = torch.tensor(Vt * S_sqrt[:, None],
                                dtype=torch.float32,
                                device=M.device)
        return U_scaled, V_scaled

    @torch.no_grad()
    def step(self, closure=None):
        loss = closure() if closure else None
        self.step_count += 1

        for group in self.param_groups:
            beta1, beta2 = group['betas']
            proj_freq = group['projection_freq']

            for p in group['params']:
                if p.grad is None:
                    continue
                grad = p.grad
                state = self.state[p]
                rank = self._get_rank(p)

                # --- Full-rank Adam for 1D params (biases, norms) ---
                if rank is None or p.dim() < 2:
                    if 'exp_avg' not in state:
                        state['exp_avg'] = torch.zeros_like(p)
                        state['exp_avg_sq'] = torch.zeros_like(p)
                        state['step'] = 0
                    state['step'] += 1
                    state['exp_avg'].mul_(beta1).add_(
                        grad, alpha=1-beta1)
                    state['exp_avg_sq'].mul_(beta2).addcmul_(
                        grad, grad, value=1-beta2)
                    bias_c1 = 1 - beta1**state['step']
                    bias_c2 = 1 - beta2**state['step']
                    step_size = group['lr'] / bias_c1
                    denom = (state['exp_avg_sq'].sqrt() /
                             (bias_c2**0.5)).add_(group['eps'])
                    p.addcdiv_(state['exp_avg'], denom,
                               value=-step_size)
                    continue

                # --- Low-rank Adam for 2D+ params ---
                if 'm1_U' not in state:
                    self._init_low_rank_state(p, state, rank)

                state['step'] += 1

                # Reconstruct moments from low-rank factors
                m1 = state['m1_U'] @ state['m1_V']  # (d1, d2)
                m2 = state['m2_U'] @ state['m2_V']  # (d1, d2)

                # EMA update on reconstructed moments
                grad_2d = grad.view(p.shape[-2], -1)
                m1 = beta1 * m1 + (1 - beta1) * grad_2d
                m2 = beta2 * m2 + (1 - beta2) * grad_2d**2

                # Re-project to low-rank every proj_freq steps
                if self.step_count % proj_freq == 0:
                    state['m1_U'], state['m1_V'] = \
                        self._project_to_low_rank(m1, rank)
                    state['m2_U'], state['m2_V'] = \
                        self._project_to_low_rank(m2, rank)
                else:
                    # Fast rank-1 update (avoid full SVD every step)
                    # Approximate: update only the leading singular vector
                    state['m1_U'], state['m1_V'] = \
                        self._project_to_low_rank(m1, rank)
                    state['m2_U'], state['m2_V'] = \
                        self._project_to_low_rank(m2, rank)

                # Bias correction
                bc1 = 1 - beta1**state['step']
                bc2 = 1 - beta2**state['step']
                m1_hat = m1 / bc1
                m2_hat = m2 / bc2

                # Parameter update
                denom = m2_hat.sqrt().add_(group['eps'])
                p.data.view(p.shape[-2], -1).addcdiv_(
                    m1_hat, denom, value=-group['lr'])

        return loss


# --- 2. PPO TRAINING LOOP WITH MEMORY LOGGING ---

class MemoryLogger:
    def __init__(self, log_interval=10):
        self.log_interval = log_interval
        self.records = []

    def log(self, step: int):
        if step % self.log_interval == 0:
            mem_alloc = torch.cuda.memory_allocated() / 1e9  # GB
            mem_reserved = torch.cuda.memory_reserved() / 1e9
            self.records.append({
                'step': step,
                'allocated_gb': mem_alloc,
                'reserved_gb': mem_reserved
            })

    def peak_