Misadjustment

The foundational stochastic gradient algorithm where misadjustment is directly proportional to step size. LMS updates coefficients using the instantaneous gradient estimate, introducing gradient noise that prevents convergence to the true Wiener solution.

• Misadjustment scales linearly with step size μ
• Smaller μ reduces steady-state error but slows convergence
• Excess MSE equals μ × trace(R) × Jmin for small μ
• Simplicity makes it the most analyzed algorithm for misadjustment trade-offs

An algorithm achieving zero misadjustment in stationary environments by using the exact deterministic least-squares solution rather than stochastic gradients. RLS recursively computes the inverse autocorrelation matrix, eliminating gradient noise entirely.

• Convergence in approximately 2N iterations where N is filter order
• Misadjustment arises only from finite-precision arithmetic
• Computational cost of O(N²) versus O(N) for LMS
• Forgetting factor λ introduces controlled misadjustment for tracking non-stationary systems

The speed at which an adaptive algorithm approaches steady-state, forming an inverse relationship with misadjustment. Faster convergence requires larger step sizes, which amplify gradient noise and increase the final excess error floor.

• LMS convergence time constant: τ ≈ 1/(2μλₐᵥₑ)
• RLS converges an order of magnitude faster than LMS
• Misadjustment-Convergence Trade-off is a fundamental limitation of gradient-based adaptation
• Practical systems must balance tracking agility against spectral purity requirements

A phenomenon where unconstrained adaptive coefficients slowly migrate from optimal values even after convergence, increasing effective misadjustment over time. Drift occurs when small perturbations accumulate due to finite-precision arithmetic or insufficient excitation.

• Common in fixed-point implementations with narrowband signals
• Leakage factor in LMS introduces a stabilizing penalty on coefficient magnitude
• Thermal variations in power amplifiers can cause genuine optimal-point drift
• Periodic retraining or burst training resets accumulated drift errors

The primary metric for quantifying misadjustment in DPD systems, expressing the residual nonlinear distortion as a ratio relative to input signal power. NMSE directly measures the excess error beyond the theoretical minimum achievable by the model structure.

• NMSE (dB) = 10·log₁₀(E[|e(n)|²] / E[|x(n)|²])
• Typical DPD targets: -35 dB to -45 dB NMSE
• Difference between training NMSE and validation NMSE reveals overfitting
• Lower NMSE correlates directly with improved ACPR and EVM performance

The broader optimization framework underlying LMS, where parameter updates use noisy gradient estimates from individual or mini-batch samples. SGD convergence theory provides the mathematical foundation for understanding misadjustment in all stochastic adaptive systems.

• Misadjustment ∝ η² × variance of gradient estimates
• Mini-batch SGD reduces gradient noise variance by factor 1/B
• Learning rate schedules can dynamically trade convergence speed for final accuracy
• Momentum terms smooth gradient estimates, reducing effective misadjustment without sacrificing convergence rate