Mathematical Foundations
1. Bandit Algorithms (UCB)
We use Upper Confidence Bound to balance exploration and exploitation.
UCB_i(t) = μ_i(t) + c · √(ln(t) / n_i(t)) + γ · D_i(t) - η · C_i(t)
- μ_i(t): Empirical Mean Reward.
- c: Confidence parameter (Exploration).
- D_i(t): Diversity Score.
- C_i(t): Computational Cost.
2. Diversity Scoring
To ensure the model learns diverse features, we penalize redundancy using Inverse Coverage.
D_i = 1 / √(1 + c_i)
- c_i: Coverage count (How many times has variable i been used?).
3. Reward Shaping
The reward signal r is not just accuracy; it is a composite of multiple objectives:
r = α · gain + β · diversity - γ · latency - δ · cost
4. Meta-Learning (Reptile)
We use the Online Reptile update rule for fast adaptation:
θ_t+1 = θ_t + α_t · (φ_i - θ_t)
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Adaptive Learning Rate: The rate α decays over time but boosts when high rewards are found.
α_t = α_0 · e^(-λt) · (1 + β · reward_t)
5. Sketch Operators
Polynomial Sketch Approximates polynomial expansion using CountSketch (Memory efficient).
poly_sketch(x, d) ≈ Σ (|S| ≤ d) c_S · Π_(i ∈ S) x_i
Tensor Sketch
Approximates the Kronecker product x₁ ⊗ x₂ using hash-based convolution.