Task Heterogeneity
Show that optimal computational depth varies across tasks.
MATHEMATICAL FOUNDATIONS / V0.1
A mathematical theory of
energy-aware computation.
LowJoule studies artificial intelligence as an optimization problem in which computational quality and physical energy expenditure are considered jointly.
CENTRAL OBJECTIVE
π* = arg maxπ E[Q(x,yπ) − λE(τπ)]
01 / FORMAL SETTING
Tasks, policies and reasoning trajectories.
Let x ∈ X denote a task sampled from a distribution D. An intelligent system produces an output y ∈ Y through a computational policy π.
x ~ D
yπ = π(x)
Q(x,y) ∈ [0,1]
yπ = π(x)
Q(x,y) ∈ [0,1]
Rather than treating inference as a fixed sequence of operations, LowJoule models computation as a trajectory.
τ = (s₀, a₀, s₁, a₁, …, sT, y)
An action may represent the execution of another neural layer, activation of an expert, generation of another token, retrieval from memory, use of an external tool, a precision change, or termination of computation.
02 / ENERGY MODEL
Computation has a physical cost.
Each computational action at performed in state st incurs an energy cost ct. The simplified trajectory energy is therefore:
E(τ) = Σt=0T ct
A complete physical accounting may additionally include memory movement, communication, accelerator overhead and cooling.
03 / MINIMAL ENERGY
Intrinsic energy complexity.
For a required quality q and reliability level 1 − δ, define the minimum expected energy necessary to solve task x:
E*(x,q,δ) = infπ { E[E(τπ) | x] : P(Q(x,yπ) ≥ q) ≥ 1 − δ }
We define the corresponding intrinsic energy complexity as:
CE(x;q,δ) = E*(x,q,δ)
Research hypothesis H1
Real task distributions exhibit substantial heterogeneity in intrinsic energy complexity.
Var[CE(x;q,δ)] > 0
04 / MARGINAL COMPUTATION
When is another computation worth its energy?
Let ΔQt represent the expected improvement in solution quality obtained by performing another computational action.
ρt = ΔQt / ct
ρ measures marginal useful computation per unit of energy. As marginal gains decrease, continuing computation eventually becomes inefficient relative to its energy cost.
Research hypothesis H2
For a non-trivial subset of inference trajectories, marginal quality gains diminish before the fixed computational budget is exhausted.
05 / OPTIMAL STOPPING
The system must learn when to stop.
At each computational state, an adaptive policy compares the value of stopping with the expected value of continuing.
Vstop(st) = E[Qt | st]
Vcontinue(st) = max E[V(st+1) − λct]
Vcontinue(st) = max E[V(st+1) − λct]
V(st) = max {Vstop(st), Vcontinue(st)}
A practical approximation is to stop when the predicted marginal quality improvement no longer compensates for its energy cost:
ΔQ̂t ≤ λct → STOP
06 / CENTRAL CONJECTURE
The LowJoule conjecture.
Under heterogeneous task complexity, diminishing marginal gains and sufficiently accurate estimates of future computational utility, an adaptive policy can reduce expected energy expenditure relative to a fixed-compute policy while satisfying a specified quality constraint.
E[Eadaptive] < E[Efixed]
Qadaptive ≥ Qfixed − ε
Qadaptive ≥ Qfixed − ε
Status
This statement is a research conjecture. It is not presented as an established theorem. LowJoule's research program aims to determine the assumptions under which such an energy advantage can be formally proven and experimentally demonstrated.
07 / EXPERIMENTAL REQUIREMENTS
What must be demonstrated?
Diminishing Returns
Measure declining marginal quality gains during inference.
Predictability
Estimate future computational utility with bounded error.
Energy Dominance
Demonstrate lower measured energy at comparable quality.