Wong's diffusion network is a stochastic, zero-input Hopfield network with a Gibbs stationary distribution over a bounded, connected continuum. Previously, logarithmic thermal annealing was demonstrated for the diffusion network and digital versions of it were studied and applied to imaging. Recently, "quantum" annealed Markov chains have garnered significant attention because of their improved performance over "pure" thermal annealing. In this note, a joint quantum and thermal version of Wong's diffusion network is described and its convergence properties are studied. Different choices for "auxiliary" functions are discussed, including those of the kinetic type previously associated with quantum annealing.
The optimization of a function V (x), xǫD, when the dimension of the space D is large and multiple local minima exist, is a computationally difficult problem. A class of stochastic algorithms, known as simulated annealing, has been developed for the case where D is countable [7]. For optimization over a bounded continuum, a diffusion network was proposed in [13] and its thermal annealing properties were established in [10] after [7]. In this note, we study a quantum version of this system that, unlike thermal annealing, modify the objective function V in a nonlinear, nonuniform way. Quantum annealing proposals in the past include those involving the Shrödinger operator with potential V [1], and those that add an auxiliary function to V that depends on ∇V (e.g., the Ising spin glass model with an external field [3]). We consider here the latter type. Generally, the intuition behind the use of an auxiliary function is to initially perform a greater breadth of search than under pure thermal annealing search.
Consider a time-inhomogeneous system described by
where the first equation is a stochastic differential equation of the Itô type,
∇ is gradient with respect to the x variables,
T ≥ 0 the deterministic thermal/temperature process, and Γ ≥ 0 the deterministic quantum parameter process.
G is such that x k t = g(u k t ) where g is a sigmoid threshold function commonly found in neural networks:
If Ṽ ≡ 0 and Σ ≡ 0 then the two relations in (1) describe a continuoustime Hopfield network with Lyapunov function V and no external inputs (easily realized as a “neural” network when V is quadratic). If Ṽ ≡ 0 and
and T > 0 is constant, then the stationary distribution of the x process is Gibbs [13]:
where Z is the partition (normalization) function. This is immediately seen by applying Itô’s rule to (1), after which the Fokker-Planck operator [9] governing the distribution p of the x process is seen to be:
where
That is, L 0 (µ) ≡ 0. Furthermore, if T (t) = T (0)/ log 2 (2 + t) (logarithmic thermal cooling), T (0) > 2M , and the global extrema of V are assumed in the interior (0, 1) n , then time-inhomogeneous process x t converges in probability to the (ground state) set that globally minimizes the objective function V [7].
If fixed T, Γ > 0, then the invariant distribution is clearly
So, if Γ = o(T ), µ Γ is like a Gibbs distribution in the sense that it tends to indicate the globally minimizing (ground) states of V as T → ∞.
3 Quantum convergence to the Gibbs invariant
In [12] (and as explained in the recent survey [3]), a quantum annealing process is considered. They show that a faster-than-logarithmic quantum cooling schedule, Γ(t) ↓ 0 as t → ∞, can be used to establish convergence to the Gibbs invariant for fixed T > 0, i.e., not to the ground states. We now prove the analogous result for the diffusion network, subject to a more rapid cooling schedule, by adapting the thermal convergence proof in [7,10]. To this end, we show how the distribution m t of x t “tracks” the distribution µ Γ(t) (note that this is obvious for all sufficiently large t if Γ reaches zero in finite time). As the proof is a more substantive variation of [7] than for pure thermal annealing of the diffusion network, we give it in greater detail here than we did in [10]. We begin by defining
Let γ Γ be the gap between 0 and the rest of the spectrum of L Γ [2]:
subject to the constraint that φ is not constant, where integration is over (0, 1) n . Equivalently,
T (∇φ) T A(∇φ)µ dx subject to φ 2 µ Γ dx = 1 and φµ Γ dx = 0.
Theorem 3.1 For any nonincreasing, differentiable quantum schedule Γ with Γ(∞) = 0 and any constant temperature T > 0:
Proof: Take
.
where the last step is integration by parts using A(0) = 0 = A(1). Thus, by the previous expression for γ(Γ(t)) (noting
Integrating in time, we get an inequality of the form z t ≤ α t + t 0 β s z s ds where α t := z 0 + t 0 2γ(Γ(s)) ds and
So by applying Gronwall’s lemma and then multiplying by 1 ≡ exp(-t 0 β r dr)/ exp(-t 0 β r dr), we get
where the last step is integration by parts (resulting in term cancellation in the numerator) and the fact that α 0 = z 0 and αt ≡ 2γ t . Now note that as t → ∞, γ(Γ(t)) → γ(Γ(∞)) := γ(0) > 0 and Γ(t) → 0, and therefore β t → -2γ(0) < 0. Thus, the numerator and denominator of the previous display both diverge as t → ∞. Applying L’Hôpital’s rule gives that żt ≤ 2γ(Γ(t)) -β t as t → ∞. where
Proof:
Completing our adaptation of the arguments in [7,10]:
Corollary 3.1 For any nonincreasing, differentiable quantum schedule Γ with Γ(∞) = 0 and any constant temperature T > 0, there is a constant K < ∞ such that for any S ⊂ (0, 1) n ,
By the previous lemma and theorem, lim
Thus, by the continuity of z t , there exists a positive constant K < ∞ such that z t ≤ K 2 for all t ≥ 0. So, by the Cauchy-Schwarz inequality,
Substituting z t ≤ K 2 completes the proof.
Note that K will depend on the parameter z 0 .
To interpret this result, note that
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