Pipelined bounded-staleness gradient coding

(1) Integration point: modify the distributed data-parallel training loop between worker backward passes and the optimizer step. Partition the training data into n shards D_i and create r replicated assignments of every shard across workers. Maintain a coordinator-side table containing, for each i, the latest gradient tensor g_i and the model-version index v_i at which it was computed. Workers pull a model snapshot, select one assigned shard, compute one minibatch gradient, and asynchronously push (i, g_i, version) to the coordinator. The coordinator launches optimizer step t whenever every shard has a gradient with age t-v_i < c; it uses the newest entry for each shard instead of waiting for all workers from the current round. (2) Pseudocode: initialize beta^0 and table[i] = none. At each worker iteration, choose an assigned shard i, compute g = grad(loss(beta_snapshot, minibatch from D_i)), and send (i,g,version_snapshot). On receipt, replace table[i] if the version is newer. At coordinator step t, if all table entries satisfy 0 <= t-version_i < c, form hat_g = mean_i(table[i].gradient), optionally weighting by shard size, apply beta <- beta - eta*hat_g, and increment t. If coverage fails, wait for a replica or increase c. (3) Compute directly from the paper's mathematics: the aggregate, delay indices s(t,i), and bounded-staleness update. Estimate empirically the second moment ||hat_g||^2, delay histogram, parameter displacement ||beta^t-beta^{version_i}||^2, and correlation between stale and current gradients on a small validation minibatch. Estimate L locally with gradient differences, L_hat = ||g_current-g_old|| / (||beta_current-beta_old||+epsilon), and reduce eta or c when L_hat^2 times the observed displacement becomes a large fraction of ||hat_g||^2. (4) First experiment: train ResNet-18 on CIFAR-10 with 8 heterogeneous workers, comparing synchronous data parallelism, ordinary gradient coding, and pipelined coding with replication r=2 and c in {2,4,8}. Inject log-normal worker delays and occasional 10-30 second stragglers. Measure wall-clock time to a fixed validation loss, final accuracy at equal wall-clock time, total gradient FLOPs, worker utilization, maximum gradient age, and the empirical error decomposition. Success means lower time-to-target loss and higher utilization than both baselines without a material final-quality loss.