The Dampened P-Controller: Iterative Refinement of a GPU Dispatch Control System

Introduction

In the iterative development of high-performance GPU-accelerated proving pipelines, the difference between a working system and an efficient one often lies in the subtle dynamics of how work is fed to the GPU. Message 3411 captures a pivotal moment in this optimization journey: the implementation of a dampened proportional controller (P-controller) for GPU dispatch scheduling. This message, written by the AI assistant in an opencode coding session, documents the reasoning, mathematical derivation, and implementation of a critical fix to a GPU pipeline that was exhibiting runaway behavior — dispatching too many synthesis jobs too quickly, flooding the allocation budget and preventing the controller from reaching a stable steady state.

The message is deceptively short — a few paragraphs of reasoning and a code edit — but it encapsulates a deep understanding of control theory applied to systems engineering, the challenges of real-time scheduling in heterogeneous compute pipelines, and the iterative nature of performance debugging. To fully appreciate this message, we must trace the chain of reasoning that led to it, examine the assumptions baked into the original design, and understand why a simple proportional gain of 1.0 was too aggressive for this particular system.

Context: The GPU Pipeline and the P-Controller Origin Story

The broader session (segment 25 of the opencode conversation) is focused on refining the GPU dispatch mechanism for the CuZK proving engine — a system that synthesizes zero-knowledge proofs and submits them to a GPU for accelerated proving. The pipeline consists of two main stages: synthesis (CPU-bound work that generates proof partitions) and GPU proving (the actual computation on the GPU). Between them sits a dispatch loop that pulls synthesized partitions and sends them to the GPU.

The problem that precipitated this message began several messages earlier. The user observed at <msg id=3389> that "the bottleneck is we don't start enough synthesis." The original dispatch loop was dispatching one partition at a time, gated by a memory budget. This meant the GPU was frequently starved — it would finish a partition and then have to wait for the next synthesis to complete before it could start new work.

The assistant responded at <msg id=3390> by implementing a P-controller — a proportional control loop that calculates a "deficit" (the gap between a target queue depth and the current number of partitions waiting for the GPU) and dispatches that many items in a burst. The formula was simple: deficit = target - gpu_work_queue.len(), then dispatch deficit items. With a target of 8 and an empty queue, this meant dispatching 8 syntheses in a single burst.

This first version (deployed as cuzk-pctrl1) was too aggressive. As the user noted at <msg id=3408>: "Spawning much too fast." The problem was that with a gain of 1.0 (P=1), the controller would dispatch the entire deficit in one shot. When the GPU completed one partition and the waiting queue was still empty (because the burst of 8 hadn't landed yet), the deficit would again be 8, triggering another burst of 8. This positive feedback loop would instantly fill all available allocation slots, preventing the controller from ever converging to a steady state where dispatches matched GPU consumption.

The User's Request: A Dampening Factor

At <msg id=3410>, the user proposed a specific dampening strategy:

"floor(min(1, deficit *0.75)) -> deficit 1,2 -> add 1, deficit 3 -> add 2; Maybe let's also add max(.., 3), expansion of 3x per consumed entry shoooild be more than enough; Currently the problem is that we nearly instantly bump into allocating all allocatable slots so with P=1 the p-controller can't stabilize"

The user's insight was twofold. First, they recognized that the proportional gain needed to be reduced from 1.0 to 0.75 — a 25% reduction in aggressiveness. Second, they identified the need for both a lower bound (dispatch at least 1 item per GPU event, to prevent starvation) and an upper bound (at most 3 items, to cap the expansion rate). This upper bound of 3 is particularly important: it limits the "expansion ratio" — how many new synthesis jobs can be started per GPU completion — to at most 3x, preventing the runaway growth that was flooding the system.

The user's formula was initially stated as floor(min(1, deficit * 0.75)), but they immediately clarified that this didn't parse correctly (since min(1, X) for any X >= 1.34 would always be 1). The corrected intent was a clamped, dampened dispatch: dispatch at least 1, at most 3, scaled by 0.75.

The Assistant's Reasoning: Decoding the Formula

Message 3411 opens with the assistant's internal reasoning, which reveals the cognitive process of interpreting the user's somewhat ambiguous mathematical expression. This reasoning section is fascinating because it shows the assistant working through multiple candidate formulas, testing each against the user's examples, and converging on the correct interpretation.

The assistant begins by restating the problem: "Currently with P=1 (gain=1), deficit=8 means we dispatch 8, which is too aggressive and fills all budget slots instantly." This establishes the baseline — the original controller had no dampening, no clamping, and no bounds.

The assistant then tries to parse the user's formula floor(min(1, deficit * 0.75)) and immediately identifies a problem: "that doesn't make sense as-is because min(1, anything >= 1.34) = 1." This is a critical moment of mathematical reasoning. The assistant recognizes that the user's written formula, taken literally, would collapse to floor(1) = 1 for any deficit >= 2, which would mean dispatching exactly 1 item regardless of how large the deficit grows. This would be a pure I-controller (integral-only) behavior — dispatching one per event — which doesn't match the user's stated examples where deficit=3 should dispatch 2.

The assistant hypothesizes that the user meant floor(max(1, deficit * 0.75)) — a floor of the max of 1 and the dampened deficit. But this still doesn't account for the upper bound of 3 that the user mentioned in their follow-up.

The reasoning then iterates through several candidate formulas:

  1. floor(max(1, min(3, deficit * 0.75))) — applies the 0.75 dampening, then clamps between 1 and 3, then floors.
  2. max(1, min(3, floor(deficit * 0.75))) — floors first, then clamps.
  3. min(floor(max(1, deficit * 0.75)), 3) — floors the max, then caps at 3.
  4. max(1, min(3, (deficit * 3) / 4)) — an integer-arithmetic variant using integer division. The assistant ultimately settles on max(1, min(3, floor(deficit * 0.75))) and verifies it against the user's examples: - deficit 1,2 → floor(10.75)=0, floor(20.75)=1 → max(1, min(3, 0))=1, max(1, min(3, 1))=1 → dispatch 1 ✓ - deficit 3 → floor(30.75)=2 → max(1, min(3, 2))=2 → dispatch 2 ✓ - deficit 4+ → floor(40.75)=3, floor(5*0.75)=3, etc. → max(1, min(3, 3))=3 → dispatch 3 ✓ This matches the user's specification exactly. The assistant then summarizes the behavior concisely: "So the dispatch formula caps out at 3 for any deficit of 4 or more, which gives us that 3x expansion ratio per GPU completion. This is essentially a damped proportional controller with a gain of 0.75, floored to integers, clamped between 1 and 3 to prevent both starvation and runaway growth."

Assumptions and Design Decisions

The assistant's reasoning reveals several important assumptions:

Assumption 1: The deficit is the right control signal. The entire P-controller approach assumes that the number of partitions waiting for the GPU (gpu_work_queue.len()) is a meaningful indicator of system pressure. This is a reasonable assumption — queue depth is a classic control signal in scheduling theory — but it has limitations. As the user later notes at <msg id=3426>, "the Waiting count is still the best signal we have from scheduling theory, but the input is too coarse." The deep synthesis pipeline introduces latency between when a dispatch decision is made and when the resulting synthesis job appears in the waiting queue, making the signal noisy and delayed.

Assumption 2: Integer dispatch counts are sufficient. The controller dispatches whole numbers of partitions — you can't dispatch 0.75 of a synthesis job. This means the controller's output is inherently quantized, which introduces granularity error. At low deficits (1-2), the floor operation rounds the dampened value down to 1, effectively giving a gain of 1.0 for those small deficits. At deficit=3, the gain is 2/3 ≈ 0.67. At deficit=4+, the gain is at most 3/deficit, approaching 0.75 asymptotically. This non-linear gain profile is a consequence of integer quantization.

Assumption 3: The upper bound of 3 is sufficient to prevent runaway growth. The user asserts that "expansion of 3x per consumed entry should be more than enough." This assumes that the system's ability to absorb new work (allocation slots, memory budget) grows more slowly than 3x per GPU completion. If the budget could absorb 4+ new slots per completion, the cap would still be the binding constraint. The assistant accepts this assumption without questioning it, which is reasonable given the user's domain expertise.

Assumption 4: The lower bound of 1 prevents starvation. By always dispatching at least 1 item per GPU event (when the queue is below target), the controller ensures the GPU never stalls completely. This is a safe assumption — even if the deficit is small, the GPU gets work. However, it also means the controller can never fully stop dispatching as long as the queue is below target, which could lead to overshoot if the gain is too high.

The Implementation: From Formula to Code

After the reasoning section, the assistant reads the source file /tmp/czk/extern/cuzk/cuzk-core/src/engine.rs to find the relevant code. The file excerpt shows the dispatch logic at lines 1275-1281, where the original P-controller checked if the waiting count was at or above the target and skipped dispatching if so.

The assistant's edit (visible in the subsequent messages at <msg id=3412>) replaces the simple deficit calculation with the dampened formula. The key change is transforming:

let deficit = target - waiting;

into something like:

let deficit = if target > 0 {
    let raw_deficit = target - waiting;
    max(1, min(3, (raw_deficit * 3) / 4))  // floor(deficit * 0.75) using integer math
} else {
    1
};

(The exact code is not shown in message 3411, but the subsequent messages confirm the edit was applied successfully and compiled cleanly.)

The assistant also updates the logging to show both the raw deficit and the clamped burst size, so the operational behavior is observable in the logs. This is visible at <msg id=3425> where the deployed binary logs: dispatch: starting burst waiting=0 target=8 deficit=8 burst=3 — confirming that a deficit of 8 is clamped down to a burst of 3.

The Deployment and Verification

The assistant's workflow after the edit is methodical and demonstrates a disciplined deployment pipeline:

  1. Compile check (cargo check at <msg id=3417>): Verifies the code compiles without errors. The only output is a pre-existing warning about JobTracker visibility, which is unrelated to the change.
  2. Docker build (docker build at <msg id=3418>): Builds a release binary inside a Docker container, ensuring a clean, reproducible build environment. The build takes 107 seconds.
  3. Binary extraction (docker create + docker cp at <msg id=3419>): Extracts the compiled binary from the Docker image to the host filesystem.
  4. SCP transfer (scp at <msg id=3420>): Copies the binary to the remote machine at /data/cuzk-pctrl2.
  5. Process management (at <msg id=3421>): Kills the running cuzk-pctrl1 process, then waits 90 seconds for the pinned memory to be freed (a known cleanup delay from the pinned memory pool work).
  6. Start new binary (nohup at <msg id=3423>): Launches the new binary with the same configuration, redirecting output to a new log file.
  7. Verification (tail at <msg id=3424>): Checks the log to confirm the new binary started successfully and the dampened dispatch logic is active. This deployment pipeline is worth noting because it reflects the production-critical nature of the system. Each iteration — from code change to running binary — takes several minutes, including a 90-second wait for memory cleanup. The assistant cannot afford to make mistakes, as each deployment cycle is costly in time.

The Result: What the Dampened Controller Achieves

The log output at <msg id=3425> confirms the dampened controller is working as designed:

dispatch: starting burst waiting=0 target=8 deficit=8 burst=3

Instead of dispatching 8 syntheses in the initial burst (which would instantly fill all budget slots), the controller dispatches only 3. As GPU events fire and the waiting queue remains below target, the controller will dispatch additional bursts of 1-3 items each time, gradually ramping up the pipeline until it reaches a steady state where dispatches match GPU consumption.

The assistant summarizes the expected behavior: "As GPU events fire and waiting stays below target, it'll ramp up with bursts of 1-3 until it stabilizes."

Limitations and the Path Forward

Despite the careful implementation, the dampened P-controller ultimately proved insufficient. At <msg id=3426>, the user reports: "Still not stable, this is a pretty deep pipeline." The fundamental issue is that the waiting count — while theoretically the best signal — is too coarse and too delayed for the deep synthesis pipeline. The latency between dispatching a synthesis job and having it appear in the GPU waiting queue introduces a phase lag that a simple P-controller cannot compensate for.

The user's proposed solution is a more sophisticated PI controller operating on a smoothed signal like an Exponential Moving Average (EMA) of the waiting count or GPU consumption rate. This is a natural evolution: the P-controller's proportional term provides responsiveness, but the integral term (I) accumulates error over time to drive the system toward the target despite persistent disturbances. The EMA smoothing addresses the noise in the raw waiting count signal.

This progression — from a reactive semaphore, to a P-controller, to a dampened P-controller, to a PI controller with EMA smoothing — mirrors the classic engineering journey of control system design. Each iteration reveals new dynamics and constraints, and the control law evolves to handle them.

Input Knowledge Required

To fully understand message 3411, the reader needs:

  1. Understanding of proportional control: The concept that a controller computes an error signal (deficit) and applies a proportional gain to determine the output. The gain of 0.75 means the controller is 25% less aggressive than the original gain of 1.0.
  2. Integer arithmetic and clamping: The formula max(1, min(3, floor(deficit * 0.75))) requires understanding how floor, min, and max compose to create a bounded, quantized output.
  3. GPU pipeline architecture: Knowledge that the system has a synthesis stage (CPU) and a proving stage (GPU), with a dispatch loop between them, and that "budget slots" represent a finite resource that limits how many synthesis jobs can be in flight simultaneously.
  4. The pinned memory pool context: The 90-second cleanup delay after killing a process is a consequence of the pinned memory pool work from earlier segments, where GPU memory allocations must be freed before a new process can start.
  5. Domain-specific terminology: Terms like "deficit," "burst," "target," "waiting count," and "allocation slots" are used with specific meanings in this system.

Output Knowledge Created

This message produces:

  1. A concrete formula for dampened dispatch: max(1, min(3, floor(deficit * 0.75))) with verified behavior for all deficit values.
  2. A code edit to engine.rs that implements the formula, changing the dispatch behavior from "dispatch all deficit" to "dispatch clamped, dampened burst."
  3. A deployed binary (cuzk-pctrl2) running on the remote machine, with observable behavior in the logs.
  4. A verified failure mode: The dampened P-controller is still not stable, confirming that the fundamental issue is not just gain tuning but the suitability of the control signal itself.
  5. A documented design iteration: The reasoning and implementation serve as a record of what was tried and why it didn't work, informing the next design (PI controller with EMA).

Mistakes and Incorrect Assumptions

While the implementation is technically correct, several assumptions proved incorrect:

The waiting count is a sufficient control signal. The deep synthesis pipeline introduces too much latency between dispatch and the appearance of work in the waiting queue. By the time the controller sees the effect of its action, the system state has already changed, leading to oscillations.

A P-controller with clamped output can stabilize the system. The clamping prevents runaway growth but doesn't address the underlying phase lag. The controller still reacts to the current deficit without accounting for the pipeline delay.

Integer quantization is acceptable. The floor operation introduces non-linearity that makes the effective gain vary with deficit size. At small deficits, the gain is effectively 1.0 (since floor(1*0.75)=0, clamped to 1), which is the same as the original aggressive controller.

The upper bound of 3 is sufficient. While it prevents the worst runaway scenarios, it also limits the controller's ability to quickly fill a deep pipeline during startup or after a long stall. The user later addresses this with a more sophisticated PI approach that doesn't need such aggressive clamping.

Conclusion

Message 3411 is a microcosm of the engineering process: a precise problem statement from the user, careful mathematical reasoning by the assistant, clean implementation, disciplined deployment, and empirical validation that reveals the next limitation. The dampened P-controller represents an important iteration in the journey from a simple reactive system to a sophisticated control loop that can handle the complex dynamics of a heterogeneous GPU pipeline.

The message also illustrates the value of the opencode session format, where the assistant's reasoning is fully visible. We can see the cognitive process of parsing an ambiguous formula, testing candidate interpretations against examples, and converging on the correct expression. This transparency is invaluable for understanding not just what was done, but why it was done that way — and what the engineer was thinking at the time.

The story doesn't end here. The dampened P-controller gives way to a PI controller with EMA smoothing in the subsequent chunk, and the journey continues. But message 3411 stands as a clear, well-documented snapshot of a critical design decision, capturing both the mathematics of control theory and the practical realities of deploying performance-critical systems in production.