Tensor Parallelism for LLM Training is a sophisticated model parallelism approach that partitions weight matrices across multiple GPUs/TPUs, enabling training of trillion-parameter language models by distributing computation and memory load.

Column and Row Parallel Linear Layers

• Tensor Parallel Concept: Weight matrices (W) split across device axis (column or row), enabling parallel matrix multiplication without replicating activations.
• Column-Parallel Linear: W divided by output dimension (Y = A × W_col, split across GPUs). Each GPU computes partial output; all-reduce aggregates results.
• Row-Parallel Linear: W divided by input dimension. Each GPU computes partial activation independently; all-gather concatenates results for next layer.
• Mixed Partitioning: Alternating column→row layers reduces synchronization overhead vs all-column. Megatron-LM uses this pattern for optimal efficiency.

Attention Head Distribution

• Multi-Head Attention Parallelism: Attention heads (H heads, typically 96-320) split across tensor-parallel devices. Each device computes subset of attention heads.
• Query/Key/Value Projection Parallelism: Q/K/V projections use column-parallel layers. Attention computation distributed across heads.
• Attention Dot-Product: Each device computes (Q × K^T) for its subset of heads independently. Softmax applied per head, values weighted locally.
• Output Projection: Multi-head outputs concatenated (all-gather), then row-parallel projection aggregates before feeding to MLP.

Megatron-LM 1D/2D/3D Tensor Parallelism

• 1D Tensor Parallelism: Splits along single dimension (typically embedding or head dimension). Simple implementation but less scalable (synchronization barrier every layer).
• 2D Tensor Parallelism: Creates 2D process grid (N_layer × N_tensor). Reduces all-reduce overhead by pipelining across two dimensions. Megatron-LM sweet spot for 100-500 GPU clusters.
• 3D Tensor Parallelism: Combines tensor parallelism with pipeline and data parallelism. Specialized for extreme scales (>1000 GPUs). Complex scheduling, minimal synchronization overhead.
• Sequence Parallelism Extension: Splits along sequence dimension (for transformer auto-regressive generation). Reduces attention O(N²) memory complexity.

All-Reduce Communication Patterns

• All-Reduce Operation: Collective communication reducing across devices (summation typical in gradient averaging). Each device sends/receives partial results.
• Ring All-Reduce: Devices arranged in logical ring. Minimizes bandwidth requirement, tolerates network asymmetry. O(NP) communication steps for N data elements, P processes.
• Tree All-Reduce: Binary tree structure reduces latency to O(log P) hops. Requires bandwidth-saturated links (not always available in over-subscribed networks).
• NCCL (NVIDIA Collective Communications Library): Optimized all-reduce kernels, automatically selects best algorithm based on hardware topology and message size.

Activation Memory and Communication Trade-offs

• Activation Recomputation: Intermediate activations dropped after forward pass, recomputed during backward pass. Reduces memory by 50% but increases computation 33%.
• Tensor Parallel Memory: No activation replicas (unlike data parallelism). Memory scales as O(model_size / tensor_parallel_degree + batch_size).
• Communication vs Computation Ratio: All-reduce bandwidth requirement ~2× (send/receive) weight size per iteration. Optimized via asynchronous communication overlap.
• Network Saturation: Bandwidth-limited at scales >100 GPUs. Network topology (fat-tree, dragonfly) critical to avoiding communication bottleneck.

Efficiency and Scaling Characteristics

• Arithmetic Intensity: Each all-reduce involves O(model_size) bandwidth for O(model_size) computation. Arithmetic intensity ~ 1 FLOP/Byte (memory-bound).
• Scaling Law: Perfect scaling requires communication hidden behind computation. Overlapping communication with matrix multiplications maintains efficiency to ~64-128 GPU clusters.
• Diminishing Returns: Beyond tensor_parallel_degree ~64, synchronization overhead dominates. Hybrid 2D/3D parallelism required for 1000+ GPU training.
• Hyperparameter Tuning: Learning rate, batch size, gradient accumulation adjusted per parallelism configuration. Different configurations yield different convergence behavior.