One Page Summary: Incremental, Iterative Processing with Timely Dataflow

This paper describes Naiad distributed computation system. Naiad uses dataflow model to represent the computations, but it aims to be a general dataflow framework in contrast to other specialized approaches such as TensorFlow. Similarly to other dataflow systems, the computations are represented as graphs, where vertices represent data and operations and edges carry the data between nodes.

Naiad was designed as the generic framework to support iterative and incremental computations with the dataflow model. We can think of an iterative computation as some function Op is executed repeatedly. Such iteration function can be looped on its output until there are no changes between the input and the output and the function converges to a fixed point.

Incremental computations are a bit more general then the iterative. In incremental processing, we start with initial input A0 and produce some output B0. At some later point, we have a change δA1 to the original input A0, such that we can have new input A1 = A0 + δA1. Incremental model produces an incremental update to the output, so δB1 = Op(δA1) and B1 = δB1 + B0. Note that incremental model only needs to have previous state (i.e. A0 and B0) to compute the next state, however we can extend it to have all output differences:. Incremental computations can be adopted for iterative algorithms where each iteration produces the difference output and next iteration operates only on that difference and not the full input. However with basic incremental computation approach it becomes impossible to do iterative operations under the changing or streaming input, as now we need to keep track not only of the iteration number (and differences between iterations), but also on the version of the input (and differences of the input).

Differential computation model overcomes the limitation of basic Iterative and Incremental approaches, by keeping all the differences δB and δA, and not just the previous state. In addition, a two-component timestamp is now used, where one component keeps track of the input version and the second value is responsible for the iteration number. Such timestamping for differences complicates the computation, as the timestamps no longer have a total order. In other words, sometimes it is not possible to tell whether one timestamp happened before another. However, the new timestamp system has a partial order for which . And under this partial order it is still possible to sum the differences together:

With differential model, we can calculate not only the end result of an operation when the new input comes in, but also any intermediate δBt. A lot of the power of differential dataflow lies with the differential operator that must produces the differences that can be summed with the equation above.

The timestamp plays a crucial role in tracking execution progress. Naiad’s communication methods Send and OnRecv can be triggered multiple time for the same variable, making it necessary to have a mechanism capable of notifying other nodes when certain data has been sent in full.  This notification triggers when all messages at or before a particular timestamp have been sent. Upon receiving a notification on a node, OnNotify(t) method is called, allowing the algorithm to react. Dataflow model complicates the notification mechanism, since the timestamps no longer have a total order, however the partial-order we have established earlier along with some dataflow graph restrictions allow Naiad to keep the effective notification system that does not break its guarantees even under nested loops and continuous input updates.