# Binomial-thinning (s,S) capacity controller - ID: 1521 - Canonical URL: https://synthcore.org/idea/1521/binomial-thinning-s-s-capacity-controller - API JSON: https://synthcore.org/api/idea/1521.json - API Markdown: https://synthcore.org/api/idea/1521.md - Verification status: unverified - Source: [arXiv:2607.28171](https://arxiv.org/abs/2607.28171) - Category: architecture - Solves: speedup, memory, stability - ML areas: moe-routing, inference-speedup, memory - Math tags: convex-analysis, stochastic-processes, optimization - Ratings: usefulness 5/10; difficulty 5/10; novelty 7/10 ## Idea description Replace continuously fluctuating conditional-computation decisions with a fixed-charge (s,S) controller for the number of active experts or channel groups. If the currently provisioned capacity falls below s, activate capacity up to S; otherwise retain the current capacity, preventing repeated small routing or kernel-launch decisions. Binomial thinning models the random subset of provisioned experts or channels that are actually available after token load, dropout, failures, or admission limits. ## Mathematical statement The key result is Lemma 6, preservation of discrete convexity under binomial thinning. A function f: Z -> R is discretely convex when its forward difference Delta f(n)=f(n+1)-f(n) is nondecreasing in n. Let xi_j be independent Bernoulli random variables with success probability gamma in [0,1], let Z^y=sum_{j=1}^y xi_j with Z^0=0, and define F(y)=E[f(Z^y)]. Since Z^(y+1)=Z^y+xi_(y+1), the proof obtains F(y+1)-F(y)=gamma E[f(Z^y+1)-f(Z^y)]=gamma E[Delta f(Z^y)], which is nondecreasing in y; hence F is discretely convex. For an ML controller, y is the number of provisioned experts or channel groups, Z^y is the random number available to process the current batch, and f(z) is the measured downstream loss plus latency or overflow cost when z units are available. Add a fixed activation cost K whenever y changes upward and a variable cost c per activated unit. The resulting fixed-charge objective has the (s,S) interpretation: below threshold s, activate up to S; at or above s, do nothing. The convexity result justifies estimating f on a discrete capacity grid and applying the stochastic availability transformation without destroying marginal-cost monotonicity. ## Key formulas - $$Z^{y}:=\sum_{j=1}^{y}\xi_{j},\qquad \xi_j\stackrel{\mathrm{i.i.d.}}{\sim}\operatorname{Bernoulli}(\gamma),\qquad Z^{y}\sim\operatorname{Binomial}(y,\gamma).$$ - $$F(y):=\mathbb{E}[f(Z^{y})],\qquad \Delta f(n):=f(n+1)-f(n),$$ - $$F(y+1)-F(y)=\gamma\,\mathbb{E}\bigl[f(Z^{y}+1)-f(Z^{y})\bigr]=\gamma\,\mathbb{E}[\Delta f(Z^{y})],$$ - $$y_t=\begin{cases}S_t,&a_ta_t\}+c(y_t-a_t).$$ ## Implementation notes Integrate the controller into a sparse MoE layer or a conditional channel layer at inference time. Let a_t be the number of currently provisioned experts, let d_t be the token demand in the next batch, and let y_t be the number provisioned after a possible activation. First collect a calibration set and measure f_t(z): mean task loss or perplexity degradation plus latency and dropped-token penalty when exactly z experts are available. Estimate gamma_t as the empirical availability rate, gamma_t=(number of expert slots that successfully accept work)/(number provisioned slots), optionally conditioned on batch size. Construct F_t(y) from the paper's formula by Monte Carlo sampling Z from Binomial(y,gamma_t) and averaging the lookup-table values f_t(Z). Enforce discrete convexity by isotonic regression on the first differences F_t(y+1)-F_t(y). Use the fixed-charge objective C_t(y|a_t)=F_t(y)+c max(y-a_t,0)+K 1[y>a_t]+h max(a_t-y,0), where h penalizes unavailable capacity or dropped tokens. For each capacity y in a small grid, compute C_t and select the minimizing target S_t; define s_t as the largest current capacity for which activating to S_t is cheaper than retaining capacity. At runtime, measure a_t and demand d_t; if a_t