Dash Faster Shampoo Optimizer

DASH: Faster Shampoo via Batched Block Preconditioning

Problem Context

The Shampoo optimizer achieves superior training performance (won MLCommons AlgoPerf competition) but suffers from significant computational overhead. Computing inverse matrix roots—its primary bottleneck—scales as O(n³), forcing infrequent preconditioner updates that degrade optimization quality. This limits practical adoption despite strong theoretical properties.

Core Concept

DASH introduces [batched block processing, numerical approximations, specialized GPU operations] to accelerate the most expensive components. By stacking preconditioner blocks into 3D tensors, DASH enables parallel GPU processing via batched operations, replacing sequential block computation with vectorized operations.

Architecture Overview

• Architectural optimization: Stack blocks into 3D tensors for batched processing
• Numerical improvements: Newton-Denman-Beavers (NDB) iteration and Chebyshev polynomial approximations as alternatives to eigenvalue decomposition
• Multi-Power-Iteration: Optimal matrix scaling with faster convergence
• Frequency boost: Enable more frequent updates without wall-clock slowdown
• Drop-in replacement: Compatible with existing Shampoo implementations

Implementation

Step 1: Organize weight matrices into blocks

Partition parameter matrices into blocks and stack them for batch processing.

# Block matrix organization
class BlockOrganizer:
    def __init__(self, block_size=256):
        self.block_size = block_size

    def partition_matrix(self, weight_matrix):
        """
        Partition weight matrix into blocks of size block_size x block_size.
        Returns list of blocks and metadata for reconstruction.
        """
        h, w = weight_matrix.shape
        blocks = []
        block_info = []

        for i in range(0, h, self.block_size):
            for j in range(0, w, self.block_size):
                block_h = min(self.block_size, h - i)
                block_w = min(self.block_size, w - j)

                block = weight_matrix[i:i+block_h, j:j+block_w]
                blocks.append(block)

                block_info.append({
                    'row_start': i, 'row_end': i + block_h,
                    'col_start': j, 'col_end': j + block_w,
                    'shape': block.shape
                })

        return blocks, block_info

    def reconstruct_matrix(self, blocks, block_info, original_shape):
        """
        Reconstruct original matrix from blocks.
        """
        h, w = original_shape
        reconstructed = torch.zeros(h, w, device=blocks[0].device)

        for block, info in zip(blocks, block_info):
            rs, re = info['row_start'], info['row_end']
            cs, info['col_start'], info['col_end']
            reconstructed[rs:re, cs:ce] = block

        return reconstructed

    def batch_blocks(self, blocks, batch_size=32):
        """
        Organize blocks into batches for parallel processing.
        """
        batches = []
        for i in range(0, len(blocks), batch_size):
            batch = torch.stack(blocks[i:i+batch_size])
            batches.append(batch)

        return batches

Step 2: Implement batched matrix root computation

Use batched GPU operations to compute inverse square roots in parallel.

# Batched matrix inverse square root
def batched_matrix_inv_sqrt(matrices_batch, method='ndb', num_iterations=10):
    """
    Compute (M^T M)^{-1/2} for a batch of matrices.

    Args:
        matrices_batch: Tensor of shape (batch_size, n, n)
        method: 'ndb' (Newton-Denman-Beavers), 'eigen', or 'cheby'
        num_iterations: Iterations for iterative methods
    """
    batch_size = matrices_batch.shape[0]
    n = matrices_batch.shape[1]

    if method == 'ndb':
        # Newton-Denman-Beavers iteration
        # More stable and faster than eigenvalue decomposition
        Y = matrices_batch.clone()  # Numerator
        Z = torch.eye(n, device=matrices_batch.device).unsqueeze(0).expand(
            batch_size, -1, -1
        )  # Denominator

        for _ in range(num_iterations):
            # NDB iteration
            Y_inv = torch.linalg.inv(Y)
            Z_inv = torch.linalg.inv(Z)

            Y_next = 0.5 * (Y + Z_inv)
            Z_next = 0.5 * (Z + Y_inv)

            Y = Y_next
            Z = Z_next

        inv_sqrt = Y  # Result

    elif method == 'cheby':
        # Chebyshev polynomial approximation
        # Fast for well-conditioned matrices
        # Compute eigenvalue bounds
        evals = torch.linalg.eigvalsh(matrices_batch)
        lambda_max = evals[..., -1]
        lambda_min = evals[..., 0]

        # Rescale to [-1, 1] interval
        center = (lambda_max + lambda_min) / 2.0
        half_width = (lambda_max - lambda_min) / 2.0

        # Chebyshev approximation: sum of Chebyshev polynomials
        inv_sqrt = torch.zeros_like(matrices_batch)

        for i in range(num_iterations):
            # Chebyshev polynomial evaluation (simplified)
            T_i = compute_chebyshev_polynomial(
                i, (matrices_batch - center) / half_width
            )
            inv_sqrt += T_i

    else:  # 'eigen'
        # Standard eigenvalue decomposition
        evals, evecs = torch.linalg.eigh(matrices_batch)
        inv_sqrt = evecs @ torch.diag_embed(1.0 / torch.sqrt(evals)) @ evecs.transpose(-2, -1)

    return inv_sqrt

Step 3: Implement multi-power iteration for scaling

Optimize matrix scaling to balance numerical stability and convergence.

# Multi-Power iteration for optimal scaling
def multi_power_iteration_scaling(matrix, num_iterations=5):
    """
    Compute optimal scaling for matrix using power iteration.
    This stabilizes subsequent root computations.
    """
    # Initialize random vector
    v = torch.randn(matrix.shape[0], 1, device=matrix.device)
    v = v / torch.norm(v)

    # Power iteration
    for _ in range(num_iterations):
        v = matrix @ v
        v = v / torch.norm(v)

    # Estimate largest eigenvalue via Rayleigh quotient
    lambda_max = (v.T @ matrix @ v) / (v.T @ v)

    # Scaling factor: normalize largest eigenvalue to 1
    scaling = 1.0 / (lambda_max + 1e-8)

    return scaling, lambda_max.item()

Step 4: Integrate into optimizer step

Create a drop-in replacement for Shampoo that uses DASH acceleration.

# DASH optimizer
class DASHShampoo(torch.optim.Optimizer):
    def __init__(
        self,
        params,
        lr=1e-3,
        eps=1e-10,
        block_size=256,
        update_freq=1,
        matrix_root_method='ndb'
    ):
        defaults = dict(
            lr=lr, eps=eps, block_size=block_size,
            update_freq=update_freq, matrix_root_method=matrix_root_method
        )
        super().__init__(params, defaults)

        self.block_organizer = BlockOrganizer(block_size=block_size)
        self.step_count = 0

    def step(self, closure=None):
        """
        Single optimization step using DASH-accelerated Shampoo.
        """
        loss = None
        if closure is not None:
            loss = closure()

        self.step_count += 1

        for group in self.param_groups:
            for p in group['params']:
                if p.grad is None:
                    continue

                grad = p.grad.data
                state = self.state[p]

                # Initialize state
                if len(state) == 0:
                    state['step'] = 0
                    state['H'] = torch.eye(
                        grad.shape[1], device=grad.device
                    ) if len(grad.shape) == 2 else grad.numel()

                state['step'] += 1

                # Update preconditioning matrix H = grad @ grad^T
                if len(grad.shape) == 2:
                    # Matrix parameter
                    grad_norm = grad / (torch.norm(grad) + group['eps'])
                    state['H'] += grad_norm @ grad_norm.T

                    # Block-wise inverse square root computation
                    if state['step'] % group['update_freq'] == 0:
                        blocks, block_info = self.block_organizer.partition_matrix(
                            state['H']
                        )
                        batch_blocks = self.block_organizer.batch_blocks(
                            blocks, batch_size=32
                        )

                        # Batched computation
                        inv_sqrt_blocks = []
                        for batch in batch_blocks:
                            inv_sqrt_batch = batched_matrix_inv_sqrt(
                                batch, method=group['matrix_root_method']
                            )
                            inv_sqrt_blocks.extend(inv_sqrt_batch)

                        # Reconstruct and apply preconditioned update
                        H_inv_sqrt = self.block_organizer.reconstruct_matrix(
                            inv_sqrt_blocks, block_info, state['H'].shape
                        )

                        # Parameter update
                        p.data -= group['lr'] * (grad @ H_inv_sqrt)

                else:
                    # Vector parameter: use diagonal approximation
                    state['H'] += grad ** 2
                    h_inv_sqrt = 1.0 / torch.sqrt(state['H'] + group['eps'])
                    p.data -= group['lr'] * grad * h_inv_sqrt

        return loss

Step 5: Benchmark and validate

Compare DASH against standard Shampoo to verify speedup and convergence.

# Benchmarking utility
def benchmark_optimizer(
    model, train_loader, optimizer_class, optimizer_kwargs,
    num_epochs=5, device='cuda'
):
    """
    Benchmark optimizer training speed and convergence.
    """
    import time

    model = model.to(device)
    optimizer = optimizer_class(model.parameters(), **optimizer_kwargs)
    criterion = torch.nn.CrossEntropyLoss()

    wall_times = []
    losses = []

    for epoch in range(num_epochs):
        epoch_start = time.time()
        epoch_loss = 0.0
        num_batches = 0

        for batch_idx, (data, target) in enumerate(train_loader):
            data, target = data.to(device), target.to(device)

            optimizer.zero_grad()
            output = model(data)
            loss = criterion(output, target)
            loss.backward()
            optimizer.step()

            epoch_loss += loss.item()
            num_batches += 1

        epoch_time = time.time() - epoch_start
        wall_times.append(epoch_time)
        avg_loss = epoch_loss / num_batches
        losses.append(avg_loss)

        print(f"Epoch {epoch + 1}: Loss={avg_loss:.4f}, Time={epoch_time:.2f}s")

    return {
        'wall_times': wall_times,
        'losses': losses,
        'total_time': sum(wall_times)
    }

Practical Guidance

When to use: Large-scale model training where preconditioner computation is a bottleneck (1B+ parameters). Most beneficial for dense, fully-connected layers.

Hyperparameters:

• Block size: 256 (typical); balance between parallelism and computation per block
• Update frequency: 1 (update preconditioner every step); increase to 2-5 for larger savings
• Matrix root method: 'ndb' (recommended default for stability), 'cheby' (fast for well-conditioned)
• Learning rate: Same as standard Shampoo; no tuning needed

Key performance metrics:

• Speedup: 4-4.83x on standard Shampoo implementation
• Wall-clock improvement: ~40-50% overhead vs. SGD (vs. 90%+ for unoptimized Shampoo)
• Convergence: Often better than SGD due to improved preconditioner estimation

Common pitfalls:

• Block size too small → excessive overhead from block management
• Block size too large → reduces parallelism
• Forgetting to use batched operations → negates acceleration benefits
• Not validating numerical stability with NDB; eigenvalue decomposition safer but slower

Scaling: Benefits scale with parameter count. Minimal benefits for small models (<100M). Optimal for dense 1B-70B models.

Reference

Paper: https://arxiv.org/abs/2602.02016 Code: Available at author's repository Related work: Shampoo optimizer, preconditioning, second-order optimization Benchmarks: Llama-953M, perplexity metrics, training curves