Determinism and Reproducibility

Automatic differentiation systems are often assumed to be deterministic. Given identical inputs, identical parameters, and identical code, many users expect identical gradients and identical optimization trajectories.

In practice, this assumption frequently fails.

Modern differentiable systems execute across:

massively parallel hardware,
distributed accelerators,
asynchronous communication networks,
mixed precision arithmetic,
optimized compiler pipelines,
stochastic training procedures.

Small numerical differences accumulate over millions or billions of operations. Two executions of the same model may therefore diverge noticeably over time.

Determinism and reproducibility are distinct concepts.

Concept	Meaning
Determinism	Same execution produces identical outputs bit-for-bit
Reproducibility	Results remain statistically or scientifically consistent

Perfect determinism is difficult in large-scale AD systems. Reproducibility is often the practical target.

Sources of Non-Determinism

Non-determinism arises from many layers of the system stack.

Source	Mechanism
Floating point addition order	Non-associativity
Parallel execution	Race-dependent reductions
GPU kernels	Undefined execution order
Atomic operations	Scheduling variation
Mixed precision	Rounding sensitivity
Random initialization	Different parameter trajectories
Compiler optimization	Reordered arithmetic
Distributed communication	Non-deterministic timing
Hardware variation	Different math implementations
Approximate kernels	Reduced numerical consistency

Automatic differentiation inherits all of these effects.

Floating Point Non-Associativity

Floating point addition satisfies:

(a+b)+c \neq a+(b+c).

This is fundamental.

Example:

(10^{20} + (-10^{20})) + 1 = 1,

while:

10^{20} + ((-10^{20}) + 1) = 0.

The mathematical expression is identical. The execution order changes the result.

Parallel gradient reductions therefore become order-sensitive.

Parallel Reductions

Reverse mode accumulates gradients:

\bar{x} \mathrel{+}= \bar{y} \frac{\partial y}{\partial x}.

On parallel hardware, accumulation may occur in different orders across runs.

A reduction tree like:

((a+b)+(c+d))

may become:

(a+(b+(c+d))).

The resulting floating point values differ slightly.

In deep optimization, these tiny perturbations may eventually produce entirely different parameter trajectories.

Chaotic Optimization Dynamics

Neural network optimization is often highly sensitive to perturbations.

Suppose two gradient trajectories differ initially by:

10^{-12}.

After millions of updates, the parameter vectors may diverge substantially.

The system behaves like a chaotic dynamical process.

Even tiny nondeterministic perturbations therefore become amplified over long training horizons.

GPU Execution Order

GPUs execute thousands of threads concurrently.

The exact scheduling order may vary depending on:

hardware occupancy,
warp scheduling,
memory timing,
compiler decisions,
concurrent workloads.

Operations involving atomics or reductions may therefore produce nondeterministic outputs.

Atomic Operations

Atomic accumulation ensures correctness but not deterministic ordering.

Suppose many threads update:

g \mathrel{+}= \delta_i.

The arrival order of updates varies.

Since floating point addition is non-associative, final gradients differ slightly across runs.

Kernel Fusion

Modern compilers aggressively fuse operations.

Example:

a \times b + c

may compile into fused multiply-add:

\operatorname{fma}(a,b,c).

FMA rounds only once instead of twice.

This changes numerical results.

Different compiler versions or hardware architectures may therefore produce different gradients.

Mixed Precision Non-Determinism

Mixed precision increases sensitivity to rounding.

Float16 and bfloat16 have coarse representational granularity.

Small perturbations that would be negligible in float64 may alter execution paths or optimizer dynamics significantly in low precision.

Loss scaling introduces additional branching behavior:

overflow detection,
scaling adjustment,
dynamic precision changes.

These mechanisms may diverge between runs.

Random Number Generation

Many differentiable systems use randomness:

parameter initialization,
dropout,
stochastic depth,
data augmentation,
reinforcement learning,
sampling-based inference.

Reproducibility requires deterministic random number generation.

However, distributed execution complicates this.

Parallel execution may consume random streams in different orders across runs.

Data Loader Non-Determinism

Training pipelines often shuffle data asynchronously.

Sources of nondeterminism include:

Source	Example
Multithreaded loading	Different batch arrival timing
Filesystem ordering	Unstable directory traversal
Distributed sharding	Worker timing differences
Augmentation pipelines	Parallel randomness

Even data ordering differences may substantially alter optimization trajectories.

Distributed Training

Distributed systems amplify nondeterminism further.

Gradient synchronization depends on:

network timing,
reduction topology,
asynchronous overlap,
communication scheduling.

All-reduce operations may aggregate gradients in different orders across runs.

Distributed optimization therefore rarely achieves bitwise reproducibility.

Compiler Transformations

AD compilers perform many graph transformations:

Optimization	Effect
Constant folding	Changes evaluation order
Algebraic simplification	Changes rounding behavior
Kernel fusion	Alters intermediate precision
Vectorization	Reorders arithmetic
Parallelization	Changes reduction structure

Mathematically equivalent programs may therefore behave numerically differently.

Control Flow Sensitivity

Branching computations can magnify tiny perturbations.

Example:

\text{if } x > 0.

A minute floating point difference may select a different branch.

The resulting computational graphs become entirely different.

This produces discontinuous optimization trajectories.

Non-Deterministic Libraries

Some numerical libraries prioritize throughput over reproducibility.

Examples include:

nondeterministic convolution algorithms,
approximate reductions,
relaxed synchronization kernels.

Deterministic alternatives often exist but may be slower.

Deterministic Modes

Many frameworks provide deterministic execution modes.

These typically:

disable nondeterministic kernels,
enforce fixed reduction ordering,
restrict parallel algorithms,
disable certain compiler optimizations.

Tradeoffs include:

Benefit	Cost
Reproducibility	Lower performance
Stable debugging	Reduced throughput
Easier verification	Higher memory usage

Bitwise Reproducibility

Bitwise reproducibility means:

x_{\text{run1}} = x_{\text{run2}}

exactly at the binary level.

This is extremely strict.

Achieving it across:

different GPUs,
different driver versions,
different compilers,
distributed systems,

is often impractical.

Statistical Reproducibility

Scientific reproducibility usually requires weaker guarantees.

Example goals:

similar validation accuracy,
consistent convergence behavior,
statistically equivalent outcomes.

Exact bitwise equality is often unnecessary.

Reproducibility in Scientific Computing

Scientific simulations frequently require stronger guarantees than machine learning.

Examples:

climate modeling,
PDE solvers,
differentiable physics,
computational chemistry.

Tiny numerical perturbations may alter long-term simulation behavior.

Deterministic numerics therefore become more important.

Reverse Mode and Determinism

Reverse mode is particularly sensitive because:

gradients accumulate from many paths,
reduction ordering matters,
backward kernels often run in parallel,
checkpoint recomputation may differ slightly,
adjoint propagation amplifies perturbations.

The backward pass may therefore be less reproducible than the forward pass.

Checkpointing Effects

Checkpoint recomputation can introduce nondeterminism.

If recomputed forward values differ slightly from original values, gradients change correspondingly.

Sources include:

stochastic operations,
random augmentation,
nondeterministic kernels,
hardware timing variation.

Deterministic checkpointing requires carefully controlled recomputation.

Numerical Drift

Suppose each operation introduces tiny perturbation:

\epsilon_i.

Over long computations:

\sum_i \epsilon_i

may become substantial.

Deep learning systems execute trillions of floating point operations during training.

Tiny drift accumulates into macroscopic differences.

Reproducibility and Optimization Basins

Optimization landscapes often contain many local minima.

Small perturbations may steer training into different basins.

Two runs may therefore:

converge differently,
generalize differently,
produce different feature representations,

despite identical architecture and data.

Seed Management

Reproducibility requires careful seed control.

Typical seeds include:

Component	Seed
Parameter initialization	RNG seed
Data shuffling	Sampler seed
Dropout	Layer seed
Augmentation	Transform seed
Distributed workers	Worker seeds

Missing even one source may break reproducibility.

Cross-Hardware Reproducibility

Different hardware may implement math differently.

Examples:

Hardware difference	Consequence
FMA behavior	Different rounding
Transcendental approximations	Slight output differences
Denormal handling	Underflow changes
Precision mode	Different accumulation

Cross-platform bitwise reproducibility is therefore extremely difficult.

Compiler and Driver Versions

Even identical hardware may behave differently under:

new compiler versions,
updated CUDA libraries,
changed drivers,
modified BLAS kernels.

Software infrastructure itself becomes part of the numerical environment.

Testing AD Systems

Deterministic execution is valuable for debugging.

Common strategies include:

Strategy	Purpose
Fixed seeds	Stable initialization
Single-thread execution	Deterministic ordering
CPU execution	Reduced parallel nondeterminism
Deterministic kernels	Stable reductions
Float64 debugging	Reduced rounding sensitivity

Many bugs become easier to isolate under deterministic execution.

Formal Verification Challenges

Nondeterminism complicates formal reasoning about AD systems.

A proof about one execution path may not hold identically across all schedules.

Parallel floating point systems therefore blur the boundary between:

numerical analysis,
compiler theory,
distributed systems,
and formal verification.

Reversible Computation

True reversibility would require recovering exact prior state.

Floating point arithmetic is not generally reversible because rounding loses information.

Thus recomputation may not exactly reproduce original values.

This complicates reversible AD systems.

Determinism vs Performance

Deterministic execution often conflicts with hardware efficiency.

High-throughput systems favor:

asynchronous execution,
relaxed synchronization,
fused kernels,
aggressive parallelism.

Deterministic execution restricts these optimizations.

The tradeoff is fundamental.

Practical Reproducibility Strategy

Large-scale systems typically aim for controlled reproducibility rather than absolute determinism.

Practical recommendations include:

Recommendation	Purpose
Fix random seeds	Stable stochastic behavior
Use deterministic kernels when debugging	Easier diagnosis
Record software versions	Environment consistency
Store model checkpoints frequently	Recovery capability
Use float64 during debugging	Reduced sensitivity
Log hardware configuration	Execution traceability
Monitor gradient statistics	Detect instability

Automatic Differentiation Perspective

Automatic differentiation itself is deterministic as a mathematical transformation.

Non-determinism enters through:

floating point arithmetic,
execution scheduling,
hardware parallelism,
compiler transformations,
stochastic algorithms.

AD systems therefore operate inside broader numerical and systems environments whose behavior may not be perfectly reproducible.

Core Idea

Determinism and reproducibility in automatic differentiation systems are limited by floating point arithmetic, parallel execution, distributed communication, stochastic optimization, and compiler transformations. Reverse-mode differentiation amplifies small numerical perturbations through large-scale gradient accumulation. Practical systems therefore balance reproducibility against performance, aiming for controlled numerical consistency rather than perfect bitwise determinism.