MobileNetV2's data-parallel forward at SYNCHRONISED BatchNorm — replica r IS shard r #

MobileNetV2FullB.lean (T2) says the typed batch-BN graph denotes mobilenetv2ForwardB_full N w on one device. MobileNetV2RenderB's data-parallel step normalises with the GLOBAL batch's statistics at replicas > 1: every one of the 52 BatchNorm sites is the sync-BN composition — this replica's mean all-reduced, then Chan's σ²_r + (μ_r − μ)² all-reduced, packed, then bnSyncF. This file is T2's data-parallel twin: that forward graph, stated as a family over the R replicas, denotes on replica r exactly batchShard r of the single-device forward at the global batch R·N.

den (mobilenetv2FwdGraphSync_full R hR N epsStr w e r)
  = batchShard R N nCls (mobilenetv2ForwardB_full (R * N) w X) r

given that each replica's input is its shard of one global batch X. ⭐ The spec does not move: the right-hand side is the committed mobilenetv2ForwardB_full, at N := R·N.

How it is proved #

As ResNet34SyncB.lean proves ResNet-34's: by induction on the chain, one block at a time, with the shard hypothesis ∀ r, den (e r) = batchShard R N _ X r carried from block to block.

• every conv, depthwise, GAP and dense node is a per-example lift (den_batchOp_shard), and so is the XLA-SAME strided conv and depthwise — the padding lives inside the per-example map;
• relu6 is pointwise, so it commutes with sharding (den_relu6_shard, the peer of den_relu_shard); the identity skip is den_addVB_shard;
• every BatchNorm site is bnSyncSiteLA, whose shard lemma den_bnSyncSiteLA is P1 on the graph read at the network index.

The graph reuses ResNet34SyncB's site verbatim — it is net-agnostic — so the only new lemma is the relu6 one; the rest of this file is MobileNetV2's six block shapes and their chain.

Names #

Parameter names are MobileNetV2FullB's (%b{k}{e,d,p}{W,g,bt}, %sW, %hW, …, the convBias := false zero-bias operands %zb{c}). A BatchNorm site with γ %b{k}dg gathers its statistics as %arsum / %armean of b{k}dgmu and b{k}dgvar, each over a [c] vector — the γ name without %, then mu / var.

What is NOT claimed here #

⚠ The backward and the parameter collectives are the T3 half (MobileNetV2SyncStepTieB.lean). ⚠ That the R replicas' inputs ARE the shards of one batch is the driver's, as in DataParallelSync.lean. ⚠ The lowerer's all_reduce is trusted as every other op's lowering is.