sec-asynchronous-distributed-minimization

12.6 Running example: asynchronous distributed minimization

Consider a global objective assembled from processor-local terms,

\[ F(x)=\sum _{i=1}^n f_i(x). \]

At an event \(e\) owned by processor \(i=o(e)\), the processor forms a local view

\[ \widehat x_e =\bigl(x_1(\rho _1(e)),\ldots ,x_n(\rho _n(e))\bigr), \qquad \rho _j(e)\in \mathord \downarrow e, \]

using the newest version of each component available in its causal past, and performs a gradient-like or stochastic update based on \(f_i\). The read maps \(\rho _j\) are part of the declaration: replacing them by the latest state in an external serialization silently gives the processor information it did not possess.

This separates two cases that look similar in scalar notation. A centralized rule that samples \(i_t\) and updates from \(f_{i_t}\) still lives on the global chain \([T]\). A distributed execution with independently generated events, delayed messages, and incomparable local histories lives on the event category \(\mathbb E\) of 12.4. Its team objective is global, but its decision sections are local. The classical bounded-delay, fairness, and contraction hypotheses are therefore assembly conditions for a global limiting section, not merely numerical tuning rules [ Bertsekas and Tsitsiklis , 1989 ] .