Kernel Implementations Analysis

Overview

The kernel implementations in ejKernel are organized by backend technology, with each backend optimized for specific hardware platforms. This modular organization allows for platform-specific optimizations while maintaining a consistent API across all implementations.

Backend Organization

ejkernel/kernels/
├── _triton/          # Triton GPU kernels (NVIDIA/AMD)
├── _pallas/          # Pallas kernels (TPU/GPU)
│   ├── tpu/         # TPU-specific implementations
│   └── gpu/         # GPU Pallas implementations
├── _xla/            # XLA-based implementations (universal)
└── _cuda/           # Direct CUDA implementations

Implementation Comparison

Backend	Target Hardware	Programming Model	Performance	Features
Triton	NVIDIA/AMD GPU	Block-based SIMD	Best on GPU	Full feature set
Pallas	TPU/GPU	Block-based	Best on TPU	Limited features
XLA	All devices	Functional	Good everywhere	Basic features
CUDA	NVIDIA GPU	Direct CUDA	Maximum control	Under development

Flash Attention Implementations

Flash Attention is the most complex and well-developed kernel, showcasing different implementation strategies across backends.

Triton Implementation

Location: ejkernel/kernels/_triton/flash_attention/_interface.py

@kernel_registry.register("flash_attention", Platform.TRITON, Backend.GPU)
@jaxtyping.jaxtyped(typechecker=beartype)
def flash_attention(
    query: Float[Array, "batch seq_len_q num_heads head_dim"],
    key: Float[Array, "batch seq_len_k num_kv_heads head_dim"],
    value: Float[Array, "batch seq_len_k num_kv_heads head_dim"],
    attention_mask: Bool[Array, "..."] | None = None,
    bias: Float[Array, "..."] | None = None,
    causal: bool = False,
    softmax_scale: float | None = None,
    dropout_prob: float = 0.0,
    dropout_seed: int | None = None,
    fwd_params: FwdParams | None = None,
    bwd_params: BwdParams | None = None,
    sliding_window: int | None = None,
    logits_soft_cap: float | None = None,
    cum_seqlens_q: Int[Array, "batch + 1"] | None = None,
    cum_seqlens_k: Int[Array, "batch + 1"] | None = None,
    softmax_aux: Float[Array, "..."] | None = None,
) -> Float[Array, "batch seq_len_q num_heads head_dim"]:
    """
    Triton-based Flash Attention v2 implementation.

    Key features:
    - Memory-efficient O(N) memory complexity
    - Online softmax computation
    - Support for variable-length sequences (cu_seqlens)
    - Logit soft capping (Gemma-2 style)
    - Custom VJP for efficient gradients
    """

Custom VJP Structure

@functools.partial(jax.custom_vjp, nondiff_argnums=(5, 6, 7, 9, 10, 13, 14))
@ejit(static_argnums=(5, 6, 7, 9, 10, 13, 14))
def flash_attention_call(query, key, value, ...):
    """Entry point with custom gradient"""
    return _fwd_attention_kernel_call(...)[0]

def _jax_fwd_attention_call(...):
    """Forward pass: computes output and saves residuals"""
    out, lse = _fwd_attention_kernel_call(...)
    residuals = (query, key, value, bias, attention_mask,
                out, lse, dropout_seed, cum_seqlens_q, cum_seqlens_k)
    return out, residuals

def _jax_bwd_attention_call(softmax_scale, dropout_prob, causal,
                            fwd_params, bwd_params, sliding_window,
                            logits_soft_cap, residuals, dO):
    """Backward pass: computes gradients"""
    query, key, value, bias, attention_mask, out, lse, ... = residuals
    dq, dk, dv = _bwd_attention_kernel_call(...)
    return dq, dk, dv, None, None, None, None, None, None

flash_attention_call.defvjp(_jax_fwd_attention_call, _jax_bwd_attention_call)

Triton Kernel Structure

def _fwd_attention_kernel_call(...):
    """Forward kernel orchestration"""
    # 1. Validate inputs
    _validate_inputs(query, key, value)

    # 2. Setup configuration
    if fwd_params is None:
        fwd_params = _get_default_fwd_params(query, key, value)

    # 3. Prepare memory layout
    block_q = fwd_params.q_blocksize
    block_k = fwd_params.kv_blocksize
    grid = calculate_grid(batch_size, num_heads, seq_len_q, block_q)

    # 4. Launch Triton kernel
    output, lse = triton_flash_attention_fwd[grid](
        query, key, value,
        bias, attention_mask,
        block_q, block_k,
        softmax_scale, causal,
        dropout_prob, dropout_seed,
        sliding_window, logits_soft_cap
    )

    return output, lse

@triton.jit
def triton_flash_attention_fwd(...):
    """Actual Triton kernel implementation"""
    # Triton block-based implementation
    # Uses shared memory for efficient data access
    # Implements online softmax algorithm
    pass

XLA Implementation

Location: ejkernel/kernels/_xla/flash_attention/_interface.py

@kernel_registry.register("flash_attention", Platform.XLA, Backend.ANY)
def flash_attention(query, key, value, ...):
    """XLA-based implementation using JAX primitives"""

Function Caching Strategy

_CORE_FUNC_CACHE = {}

def _make_core_func(precision_code, logits_dtype_code, chunk_size_q,
                   chunk_size_k, normalize_output, causal, dropout_prob):
    """
    Creates specialized function for static parameters.
    Caching prevents recompilation for common configurations.
    """
    @jax.custom_vjp
    def _flash_attention_core_specialized(query, key, value, ...):
        return _flash_attention_fwd(...)

    def _fwd(query, key, value, ...):
        y = _flash_attention_fwd(...)
        ctx = (bias, attention_mask, softmax_aux, sliding_window,
              softmax_scale, logits_soft_cap, chunk_size_q, chunk_size_k,
              normalize_output, query, key, value, causal,
              dropout_prob, dropout_key)
        return y, ctx

    def _bwd(ctx, g):
        # Unpack context
        bias, attention_mask, ... = ctx
        # Compute gradients
        dq, dk, dv = _flash_attention_bwd(...)
        return (dq, dk, dv, None, None, None, None, None, None, None)

    _flash_attention_core_specialized.defvjp(_fwd, _bwd)
    return _flash_attention_core_specialized

def flash_attention(...):
    """Main entry point"""
    # Create cache key from static parameters
    cache_key = (precision_code, logits_dtype_code, chunk_size_q,
                chunk_size_k, normalize_output, causal, dropout_prob)

    # Get or create specialized function
    if cache_key not in _CORE_FUNC_CACHE:
        _CORE_FUNC_CACHE[cache_key] = _make_core_func(*cache_key)

    # Execute specialized function
    return _CORE_FUNC_CACHE[cache_key](query, key, value, ...)

XLA Algorithm Implementation

def _flash_attention_fwd(query, key, value, ...):
    """
    XLA Flash Attention using JAX operations.

    Algorithm:
    1. Chunk Q, K, V into blocks
    2. For each Q block:
       a. Compute attention with all K, V blocks
       b. Accumulate using log-sum-exp for numerical stability
    3. Normalize final output
    """
    # Chunk inputs
    q_chunks = chunk(query, chunk_size_q, axis=1)
    k_chunks = chunk(key, chunk_size_k, axis=1)
    v_chunks = chunk(value, chunk_size_k, axis=1)

    # Process chunks with scan
    def process_q_chunk(q_chunk):
        def process_kv_chunk(acc, kv_chunk):
            k_chunk, v_chunk = kv_chunk
            # Compute attention scores
            scores = jnp.einsum("...qhd,...khd->...qhk", q_chunk, k_chunk)
            scores = scores * softmax_scale

            # Apply causal mask if needed
            if causal:
                scores = apply_causal_mask(scores)

            # Softmax
            scores = jax.nn.softmax(scores, axis=-1)

            # Apply dropout
            if dropout_prob > 0:
                scores = apply_dropout(scores, dropout_prob, dropout_key)

            # Weighted sum
            output = jnp.einsum("...qhk,...khd->...qhd", scores, v_chunk)

            # Accumulate with log-sum-exp
            return update_accumulator(acc, output, scores)

        # Scan over KV chunks
        init_acc = initialize_accumulator()
        final_acc, _ = jax.lax.scan(process_kv_chunk, init_acc,
                                    (k_chunks, v_chunks))

        return normalize_output(final_acc)

    # Map over Q chunks
    outputs = jax.vmap(process_q_chunk)(q_chunks)
    return reshape_output(outputs)

Pallas TPU Implementation

Location: ejkernel/kernels/_pallas/tpu/flash_attention/_interface.py

@kernel_registry.register("flash_attention", Platform.PALLAS, Backend.TPU)
def flash_attention(query, key, value, ...):
    """Pallas TPU implementation with TPU-specific optimizations"""

TPU-Specific Adaptations

def flash_attention_tpu(query, key, value, ...):
    """
    TPU-specific implementation differences:
    1. Uses segment IDs instead of attention masks
    2. Different block size requirements (multiples of 128)
    3. No support for certain features (sliding window, soft cap)
    """
    # TPU limitations
    if cum_seqlens_q is not None:
        raise NotImplementedError(
            "Variable-length sequences not supported on TPU"
        )
    if sliding_window is not None:
        raise NotImplementedError(
            "Sliding window not supported on TPU"
        )
    if logits_soft_cap is not None:
        raise NotImplementedError(
            "Logits soft cap not supported on TPU"
        )

    # Convert attention mask to segment IDs (TPU optimization)
    if attention_mask is not None:
        from ejkernel.xla_utils import mask_to_segment_ids
        q_segment_ids, kv_segment_ids = mask_to_segment_ids(attention_mask)
    else:
        q_segment_ids = kv_segment_ids = None

    # TPU block configuration
    block_sizes = BlockSizes(
        block_q=fwd_params.q_blocksize or 256,      # Larger blocks on TPU
        block_k_major=fwd_params.kv_blocksize or 256,
        block_k=128,  # Fixed for TPU MXU efficiency
        block_b=1,
        block_h=num_heads,
        block_q_dkv=fwd_params.q_blocksize or 256,
        block_k_dkv=fwd_params.kv_blocksize or 256,
        block_k_major_dkv=fwd_params.kv_blocksize or 256,
        block_k_dkv=128,
        block_q_dq=bwd_params.q_blocksize or 256,
        block_k_dq=bwd_params.kv_blocksize or 256,
    )

    # Use Pallas kernel
    return pallas_flash_attention(
        query, key, value,
        segment_ids=SegmentIds(q=q_segment_ids, kv=kv_segment_ids),
        block_sizes=block_sizes,
        causal=causal,
        softmax_scale=softmax_scale
    )

Pallas Kernel Definition

@functools.partial(
    jax.pallas.pallas_call,
    out_shape=compute_output_shape,
    grid=compute_grid,
    debug=False,
)
def pallas_flash_attention_kernel(
    q_ref,      # Input refs
    k_ref,
    v_ref,
    segment_ids_ref,
    o_ref,      # Output ref
    lse_ref,    # Log-sum-exp ref
    block_sizes,
    causal,
    softmax_scale,
):
    """
    Pallas kernel implementation for TPU.

    Uses TPU-specific features:
    - Matrix Multiply Units (MXU)
    - High-bandwidth memory (HBM)
    - Efficient segment ID handling
    """
    # Get block indices
    block_q_idx = jax.pallas.program_id(0)
    block_b_idx = jax.pallas.program_id(1)

    # Load Q block
    q_block = q_ref[block_b_idx, block_q_idx, :, :]

    # Initialize accumulator
    acc = jnp.zeros_like(o_ref[block_b_idx, block_q_idx, :, :])
    lse = jnp.full((block_sizes.block_q,), -jnp.inf)

    # Iterate over K, V blocks
    for block_k_idx in range(num_k_blocks):
        # Load K, V blocks
        k_block = k_ref[block_b_idx, block_k_idx, :, :]
        v_block = v_ref[block_b_idx, block_k_idx, :, :]

        # Compute attention scores using TPU MXU
        scores = jax.lax.dot_general(
            q_block, k_block.T,
            (((2,), (2,)), ((), ())),
            preferred_element_type=jnp.float32
        )

        # Scale scores
        scores = scores * softmax_scale

        # Apply causal mask
        if causal:
            mask = compute_causal_mask(block_q_idx, block_k_idx)
            scores = jnp.where(mask, scores, -jnp.inf)

        # Check segment IDs (TPU-specific)
        if segment_ids_ref is not None:
            q_segments = segment_ids_ref.q[block_b_idx, block_q_idx]
            kv_segments = segment_ids_ref.kv[block_b_idx, block_k_idx]
            segment_mask = (q_segments[:, None] == kv_segments[None, :])
            scores = jnp.where(segment_mask, scores, -jnp.inf)

        # Online softmax with log-sum-exp
        scores_max = jnp.max(scores, axis=-1, keepdims=True)
        scores_exp = jnp.exp(scores - scores_max)
        scores_sum = jnp.sum(scores_exp, axis=-1, keepdims=True)

        # Update accumulator
        prev_scale = jnp.exp(lse - scores_max)
        acc = acc * prev_scale + jax.lax.dot_general(
            scores_exp, v_block,
            (((1,), (0,)), ((), ())),
            preferred_element_type=jnp.float32
        )

        # Update log-sum-exp
        lse = jnp.logaddexp(lse, scores_max + jnp.log(scores_sum))

    # Normalize output
    o_ref[block_b_idx, block_q_idx, :, :] = acc / jnp.exp(lse)
    lse_ref[block_b_idx, block_q_idx, :] = lse

Other Attention Mechanisms

Page Attention

Optimized for KV-cache in inference scenarios with paged memory management.

@kernel_registry.register("page_attention", Platform.TRITON, Backend.GPU)
def page_attention_triton(
    query: Float[Array, "batch num_heads head_dim"],
    key_cache: Float[Array, "num_blocks num_heads block_size head_dim"],
    value_cache: Float[Array, "num_blocks num_heads block_size head_dim"],
    context_lens: Int[Array, "batch"],
    block_tables: Int[Array, "batch max_blocks_per_seq"],
    softmax_scale: float,
    block_size: int,
):
    """
    Paged attention for efficient KV-cache management.

    Features:
    - Block-wise memory management
    - Dynamic context lengths
    - Efficient cache reuse
    """

Ring Attention

Distributed attention for sequence parallelism across devices.

@kernel_registry.register("ring_attention", Platform.XLA, Backend.ANY)
def ring_attention_xla(
    query: Float[Array, "batch seq_len num_heads head_dim"],
    key: Float[Array, "batch seq_len num_heads head_dim"],
    value: Float[Array, "batch seq_len num_heads head_dim"],
    axis_name: str = "ring",
    causal: bool = True,
):
    """
    Ring attention for distributed computation.

    Algorithm:
    1. Split Q, K, V across devices
    2. Rotate K, V chunks in ring pattern
    3. Compute local attention at each step
    4. Accumulate results with proper normalization
    """
    # Get ring communicator
    axis_index = jax.lax.axis_index(axis_name)
    axis_size = jax.lax.psum(1, axis_name)

    # Initialize accumulator
    output = jnp.zeros_like(query)
    lse = jnp.full(query.shape[:-1], -jnp.inf)

    # Ring iterations
    for step in range(axis_size):
        # Rotate K, V
        k_chunk = jax.lax.ppermute(
            key, axis_name,
            perm=[(i, (i + step) % axis_size) for i in range(axis_size)]
        )
        v_chunk = jax.lax.ppermute(
            value, axis_name,
            perm=[(i, (i + step) % axis_size) for i in range(axis_size)]
        )

        # Compute local attention
        local_out, local_lse = compute_attention(
            query, k_chunk, v_chunk,
            causal=causal and (step == 0)
        )

        # Accumulate with log-sum-exp
        output, lse = accumulate_attention(output, lse, local_out, local_lse)

    return output

Block Sparse Attention

Efficient sparse patterns for long-context processing.

@kernel_registry.register("blocksparse_attention", Platform.TRITON, Backend.GPU)
def blocksparse_attention_triton(
    query: Float[Array, "batch seq_len num_heads head_dim"],
    key: Float[Array, "batch seq_len num_heads head_dim"],
    value: Float[Array, "batch seq_len num_heads head_dim"],
    sparsity_mask: Bool[Array, "num_blocks_q num_blocks_k"],
    block_size: int = 64,
):
    """
    Block-sparse attention with configurable sparsity patterns.

    Supports:
    - Fixed sparsity patterns (local, strided, etc.)
    - Learned sparsity patterns
    - Combination patterns (local + global)
    """

Non-Attention Operations

Recurrent Kernels

@kernel_registry.register("recurrent", Platform.TRITON, Backend.GPU)
def recurrent_triton(
    inputs: Float[Array, "batch seq_len input_dim"],
    hidden: Float[Array, "batch hidden_dim"],
    weights: dict[str, Array],
    cell_type: str = "lstm",
):
    """
    Optimized recurrent operations.

    Supports:
    - LSTM, GRU, vanilla RNN
    - Bidirectional processing
    - Custom activation functions
    """

Mean Pooling

@kernel_registry.register("mean_pooling", Platform.XLA, Backend.ANY)
def mean_pooling_xla(
    inputs: Float[Array, "batch seq_len dim"],
    mask: Bool[Array, "batch seq_len"] | None = None,
):
    """
    Variable-length sequence pooling.

    Features:
    - Proper masking for padded sequences
    - Numerically stable averaging
    - Custom gradients for efficiency
    """
    if mask is not None:
        # Masked mean
        masked_inputs = inputs * mask[:, :, None]
        sum_inputs = jnp.sum(masked_inputs, axis=1)
        count = jnp.sum(mask, axis=1, keepdims=True)
        return sum_inputs / jnp.maximum(count, 1.0)
    else:
        # Standard mean
        return jnp.mean(inputs, axis=1)

Grouped Matrix Multiplication

@kernel_registry.register("grouped_matmul", Platform.PALLAS, Backend.TPU)
def grouped_matmul_pallas(
    a: Float[Array, "batch groups m k"],
    b: Float[Array, "batch groups k n"],
):
    """
    Efficient batched matrix multiplication.

    Optimizations:
    - Fused operations for multiple groups
    - Efficient memory access patterns
    - TPU MXU utilization
    """
    return jax.vmap(jax.vmap(jnp.dot, in_axes=0), in_axes=0)(a, b)

Implementation Patterns

1. Custom VJP Pattern

All performance-critical kernels implement custom VJP:

@jax.custom_vjp
def kernel(inputs, params):
    return forward(inputs, params)

def kernel_fwd(inputs, params):
    output = forward(inputs, params)
    residuals = (inputs, params, intermediate_state)
    return output, residuals

def kernel_bwd(residuals, grad_output):
    inputs, params, intermediate = residuals
    grad_inputs = backward_inputs(grad_output, intermediate)
    grad_params = backward_params(grad_output, intermediate)
    return grad_inputs, grad_params

kernel.defvjp(kernel_fwd, kernel_bwd)

2. Static Parameter Caching

For XLA implementations, cache specialized functions:

_CACHE = {}

def get_specialized_func(static_params):
    if static_params not in _CACHE:
        _CACHE[static_params] = create_specialized(static_params)
    return _CACHE[static_params]

3. Platform-Specific Adaptations

def kernel(inputs, **kwargs):
    platform = detect_platform()

    if platform == "tpu":
        # TPU-specific preprocessing
        kwargs = adapt_for_tpu(kwargs)

    if platform == "gpu":
        # GPU-specific optimizations
        kwargs = optimize_for_gpu(kwargs)

    return execute_kernel(inputs, **kwargs)

4. Block Size Selection

def select_block_sizes(seq_len, head_dim, platform):
    """Heuristic block size selection"""
    if platform == "gpu":
        if head_dim <= 64:
            block_q = 128
            block_k = 256
        else:
            block_q = 64
            block_k = 128
    elif platform == "tpu":
        # TPU prefers larger blocks
        block_q = 256
        block_k = 512
    else:
        # Conservative defaults
        block_q = 32
        block_k = 64

    return block_q, block_k

Performance Characteristics

Triton (GPU)

Strengths:

• Direct control over shared memory

• Efficient warp-level operations

• Flexible block sizes

• Custom data types

Limitations:

• NVIDIA/AMD specific

• Compilation overhead

• Version compatibility

Pallas (TPU)

Strengths:

• TPU MXU utilization

• Efficient for large blocks

• Native bfloat16 support

• High memory bandwidth

Limitations:

• Fixed block size constraints

• Limited feature set

• TPU-specific code

XLA (Universal)

Strengths:

• Works on all platforms

• JAX integration

• Automatic optimization

• Good baseline performance

Limitations:

• Less control over low-level details

• May not achieve peak performance

• Limited by XLA compiler capabilities

Testing Strategy

Cross-Backend Validation

def test_attention_consistency():
    """Ensure all backends produce equivalent results"""
    # Generate test inputs
    inputs = create_test_inputs()

    # Get all implementations
    triton_impl = kernel_registry.get("attention", Platform.TRITON)
    pallas_impl = kernel_registry.get("attention", Platform.PALLAS)
    xla_impl = kernel_registry.get("attention", Platform.XLA)

    # Compute outputs
    out_triton = triton_impl(**inputs)
    out_pallas = pallas_impl(**inputs)
    out_xla = xla_impl(**inputs)

    # Compare
    assert_allclose(out_triton, out_xla, rtol=1e-5)
    assert_allclose(out_pallas, out_xla, rtol=1e-5)

Gradient Verification

def test_custom_vjp():
    """Verify custom gradients match autograd"""
    # Create test case
    inputs = create_differentiable_inputs()

    # Custom VJP gradient
    custom_grads = jax.grad(kernel_with_custom_vjp)(inputs)

    # Autograd gradient (reference)
    auto_grads = jax.grad(kernel_without_vjp)(inputs)

    # Compare
    assert_allclose(custom_grads, auto_grads, rtol=1e-5)

Conclusion

The kernel implementations in ejKernel demonstrate:

1. Platform Optimization: Each backend leverages platform-specific features

2. Consistent API: Same interface across all implementations

3. Performance Focus: Custom VJP, caching, and optimizations

4. Extensibility: Clear patterns for adding new kernels

5. Robustness: Comprehensive testing and validation

This multi-backend approach ensures optimal performance across diverse hardware while maintaining ease of use and reliability.