Restart-Based Geometric Decay

• Restart-based geometric decay is a universal mechanism where stochastic restarts force exponential (geometric) decay in survival and convergence metrics across diverse processes.
• It transforms heavy-tailed or non-ergodic dynamics into systems with strict exponential convergence by employing renewal theory and explicit Laplace or generating function solutions.
• The approach underpins accelerated convergence in optimization algorithms and ensures exponential ergodicity in Markov processes, enhancing robustness and predictability in system performance.

Restart-based geometric decay refers to the universal phenomenon whereby stochastically restarting a process at random (often memoryless) times enforces exponential (i.e., geometric) decay in survival, first-pass, convergence, or error statistics—regardless of the original process's persistence, mixing, or tail behavior. The mechanism arises in Markov processes, random walks, stochastic search, optimization algorithms, and fractional kinetics, where the insertion of an external (often Poissonian or geometric) restart schedule transforms dynamics with possibly heavy or power-law tails into systems exhibiting strict exponential convergence rates, ergodicity, or stationary heavy tails. The effect may extend to functionals, moments, and in some cases, to nontrivial coupled systems beyond simple Markovian settings.

1. Fundamental Mechanism: Renewal with Restart

Let $T$ be the random completion or absorption time of a base process (e.g., first passage, survival, or error to threshold), with survival law $S(t) = \Pr(T > t)$ or, in discrete-time, $S(n)$. Introducing restarts at random times (distributed according to a restart kernel) yields a renewal process: at each (independent) restart, the process is rejuvenated. For memoryless restarts—continuous-time Poisson ($r$), or discrete-time geometric ($q$)—the survival law admits explicit renewal equations:

• Continuous-time (Poisson restart at rate $r$):

$$S_r(t) = e^{-r t} S(t) + \int_0^t r e^{-r\tau} S(\tau)\, S_r(t-\tau)\,d\tau$$

• Discrete-time (geometric restart probability $q$):

$S_R(z) = \frac{S((1-q)z)}{1 - qz S((1-q)z)}$

The Laplace or generating function solutions feature a simple pole on the negative real axis, whose position dictates the global exponential decay rate under restart (Jr. et al., 2019, Bonomo et al., 2021).

2. Emergence of Exponential Tails and Decay Rates

For arbitrary survival laws $S(t)$ or first-passage laws $S(t) = \Pr(T > t)$0, stochastic restarts "force" the tail to become exponential (geometric in discrete-time). The decay rate $S(t) = \Pr(T > t)$1 (continuous) or $S(t) = \Pr(T > t)$2 (discrete) is the unique root of the pole in the transformed renewal equation:

• Continuous:

$S(t) = \Pr(T > t)$3

• Discrete:

$S(t) = \Pr(T > t)$4

Explicit mean and higher-moment formulas result from renewal theory, e.g. for geometric restart,

$S(t) = \Pr(T > t)$5

where $S(t) = \Pr(T > t)$6 is the generating function of first-passage times (Bonomo et al., 2021).

3. Markov Processes, Stationarity, and Ergodicity

When Poissonian restarts are introduced into continuous-time Markov processes, the resulting process is always exponentially ergodic—the total variation distance from stationarity decays exactly at rate $S(t) = \Pr(T > t)$7, the restart rate (Avrachenkov et al., 2012). The invariant measure has explicit resolvent form. Importantly, moments of the process (e.g., in geometric Brownian motion under resetting) converge exponentially fast, provided growth rates of the original moments are sufficiently controlled. This results in positive Harris recurrence and explicit geometric contraction for a wide class of systems.

Process Class	Restart Kernel	Resulting Decay
Markov, general	Poisson ($S(t) = \Pr(T > t)$8)	Exponential ($S(t) = \Pr(T > t)$9)
Lattice random walks	Geometric ($S(n)$0)	Geometric ($S(n)$1)
Fractional diffusions	Poisson	Exponential in limit, sub- or stretched exp. with coupling

4. Applications to Optimization: Geometric Error Contraction

Restart-based geometric decay is foundational in obtaining linear convergence rates in first-order and coordinate descent methods in convex optimization. In accelerated gradient or coordinate descent, scheduled restarts (either at fixed intervals or adaptively triggered) transform sublinear $S(n)$2 convergence into genuine linear (geometric) error decay, where the contraction factor per restart block can be tuned according to strong convexity, sharpness, or local error bounds (Roulet et al., 2017, Fercoq et al., 2018, Luo et al., 14 Jan 2025). The mechanism is robust: even without precise knowledge of error parameters, grid-search or adaptive schedules ensure geometric contraction, and restart does not require tuning to the exact strong convexity constant to guarantee optimal rates.

• Under strong convexity or sharpness, for error measure $S(n)$3:

$S(n)$4

where $S(n)$5 is the number of restarts.

5. Memoryless Versus Coupled and Heavy-Tailed Restarts

The imposition of geometric decay is intimately linked to the memorylessness of the restart kernel. For non-memoryless (e.g., Zeta-distributed or heavy-tailed) restart intervals, the forced exponential tail can be destroyed, and broad-tailed, non-Markovian decay profiles emerge, determined by the pole structure (or absence thereof) of the renewal equation (Bonomo et al., 2021, Jr. et al., 2019). In processes where microscopic coupling interweaves the reset with intrinsic dynamics (e.g., in subdiffusive CTRWs with decay), the decay rate becomes age-dependent, and the survival law departs from a pure exponential towards a non-Markovian, history-dependent functional form governed by a time-dependent kernel, erasing the geometric law (Jr. et al., 2019).

6. Stochastic Resetting in Physical and Financial Models

Restart-based geometric decay is central to stochastic modeling in several domains:

• In geometric Brownian motion under resetting, rare excursions lead to stationary heavy-tailed (power-law) distributions, but the sample mean and variance split into quenched, unstable, and stable annealed regimes, depending on the reset rate (Stojkoski et al., 2021). An optimal resetting rate minimizes the time required for self-averaging, i.e., for ensemble means to approximate typical samples—a quantifier of "ergodicity emergence" in nonergodic systems.
• In random walks (e.g., Pólya, Sisyphus walks), geometric restart converts first-passage and survival distributions from heavy or fat tails to pure geometric/exponential decay, simplifying both theoretical analysis and numerical computation (Bonomo et al., 2021).
• In optimization and information retrieval, restart induces robust convergence by repeatedly resetting to a known or favorable configuration, nullifying pathologies of slow mixing or heavy-tailed error.

7. Generalizations and Unified Renewal Framework

The geometric decay effect isolates the essential analytic property for universal exponential convergence: the presence of a (unique) solution to $S(n)$6 (discrete, with $S(n)$7 the appropriate generating function), or similar Laplace or resolvent relations (continuous). The framework accommodates arbitrary restart distributions, with memoryless kernels yielding the sharpest geometric decay. For functionals of renewal periods and for processes with coupled reset–process dynamics, explicit unified renewal equations determine stationary values and asymptotic behavior, encompassing moments, power spectra, and nonlocal functionals (Jr. et al., 2019).

In summary, restart-based geometric decay is a universal principle underpinning the emergence of exponential tails, linear error contraction, and explicit ergodicity in stochastic and deterministic systems subjected to memoryless or well-behaved restart schedules. It is characterized by explicit renewal-theoretic formulas that quantitatively dictate the decay rates, stationary distributions, and moment convergence under restart across stochastic processes, random walks, and optimization algorithms.

Key References:

• (Jr. et al., 2019) (Lapeyre & Dentz, Unified renewal approach)
• (Bonomo et al., 2021) (First-passage under geometric restart in lattice walks)
• (Avrachenkov et al., 2012) (Markov processes with Poisson restart)
• (Fercoq et al., 2018, Luo et al., 14 Jan 2025, Roulet et al., 2017) (Restart in accelerated and coordinate descent methods)
• (Stojkoski et al., 2021) (Geometric Brownian motion under stochastic resetting)