Nagaev Bounds for Nonconvex Momentum: Finite Moments and Temporal Dependence

Momentum can smooth individual gradient errors without removing the confidence cost of rare shocks along an optimization trajectory. For ordinary nonconvex momentum under conditional second and $p$th moments, $p>2$, we prove a bound on average squared-gradient stationarity that separates a logarithmic variance term from a polynomial rare-shock term. A quadratic construction matches the polynomial component in its powers of horizon, confidence and stepsize over the stated moment class, allowing the noise law to depend on horizon and confidence. Both bounds have constants uniform in momentum within an explicit stepsize region. We then replace conditional moment envelopes by a budget for accumulated noise energy. This gives $T^{-1/2}$ stationarity at fixed momentum and confidence under a uniform second-moment bound, including state-dependent martingale noise with unbounded conditional variance. A predictable correction extends the analysis to additive causal noise, covering polynomial linear filters with exponent $a>3/2$ under finite $p>2$ moments. Exact calculations and simulations explain the common mechanism: pointwise smoothing can coexist with a persistent normalized shock response. They also test fixed-law tails and the scope of the random-volatility and causal extensions.