文章探讨了在固定决策情境下，增加思考时间或计算资源对决策质量的提升效果。作者指出，思考的边际回报可能遵循对数规律，即初期投入显著提升质量，但随着时间增加，每单位投入的回报逐渐递减。文章分析了简单重复采样、搜索深度、混沌系统建模和组合搜索等模型，展示了不同情境下思考深度的边际效益。作者认为，除非有指数级增长的输入，否则思考深度存在实际上限，而并行计算可能无法超越串行计算的回报上限。

📈 文章的核心观点是思考的边际回报可能遵循对数规律，初期投入显著提升决策质量，但随着时间增加，每单位投入的回报逐渐递减。

🔍 作者通过简单重复采样模型指出，即使能无误差地选择最佳方案，该策略的期望值增长也受对数限制，对于正态分布的提案生成器，增长速度为对数的平方根。

🌳 在搜索深度模型中，作者认为如果收益在深度上大致均匀，则搜索的边际回报将对数增长。优秀的启发式算法可以修剪树的部分分支，但搜索空间仍呈指数级增长。

🌪️ 对于混沌系统，由于对初始条件的敏感性，预测精度提升需要指数级更精确的初始条件，这暗示了计算资源的边际效益可能受限。

🧪 文章还讨论了组合搜索，其中计划组件数量的指数级搜索空间导致收益与组件数量成正比时，搜索长度的边际回报呈对数增长。

Published on July 30, 2025 12:50 AM GMT

Lots of phenomena turn out to have logarithmic returns: to get an improvement, you double effort or resources put in, but then to get the same improvement you have to double inputs again and again and so on. Equivalently, input costs are exponential in output quality^[1]. You can probably think of some examples.

I want to know: is 'extra reasoning compute' like this? (Or, under what conditions and by what means can you beat this?) I'm especially interested in this question as applied to deliberate exploration and experiment design.

Said another way, from a given decision-context, without making extra observations or gathering extra data, what are the optimal marginal returns to 'thinking harder'^[2] about what to do next?

Intuitively, if I have a second to come up with a plan, it might be weak, five minutes and it might be somewhat reasonable, a day and it'll be better, a year (full time!) and I've reached very diminishing returns. Presumably a century in my ivory tower would be barely better. I'd usually do better trying to get more data.

Is this even a sensible question, or is 'improvement in reasoning output' far too vague to get traction here?

That's the question; below some first thoughts toward an answer.

Simple model: repeated sampling/best of k

If you have a proposal generator, and you can choose between proposals, a simple approach to getting better generations is:

sample a large number, $k$, of proposals(try to) evaluate and pick the best one

(This is actually the best strategy you could take if you can only add parallel compute, but there might be strictly better approaches if you can add serial^[3].)

Even assuming you can unerringly pick the best one, this strategy turns out to have logarithmically-bounded expected value for many underlying distributions of proposals^[4]. In fact, for a normally-distributed proposal generator, you even get the slightly worse square root of logarithmic growth^[5].

You can in principle sidestep this if your proposal generator has sufficiently heavy-tailed proposal distribution, and you can reliably ex ante distinguish better from worse at the tails.

Other rougher gestures

Search depth

Often to find approximate solutions to problems, we might employ search over a tree-like structure. This emerges very naturally for planning over time, for example, where branching options (whether choice or chance) at each chosen time interval give rise to a tree of possible plans.

If gains are roughly uniform in depth, this gives rise to logarithmic returns to further search. With excellent heuristics, you might be able to prune large fractions of the tree - this gives you a kinder exponent, but still an exponential space to search.

But why/when are gains roughly uniform in depth?

Modelling chaos

Chaotic systems are characterised by sensitivity to initial conditions: dynamics where measurement or specification errors compound exponentially.

So, to forecast at a given precision and probabilistic resolution, it takes exponentially tighter initial specification precision to forecast marginal incremental time depth. (This is why in practice we only ever successfully forecast chaotic systems like weather at quite coarse precision or short horizon.)

Specification precision doesn't exactly map to having extra compute, but it feels close. And marginal incremental time depth doesn't necessarily correspond linearly to 'goodness of plan'.

Combinatorial search

If there's some space of ingredients or components which can be combined for possible insight, the size of the search space is exponential^[6] in the number of components in the plan. So if gains are proportional to the number of components in a good plan, you get logarithmic returns to searching longer.

Something similar applies if the design possibilities benefit from combining already-discovered structures in a hierarchy.

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1. Notably this means that, unless you have an exponentially growing source of inputs to counteract it, there's a practical upper limit to growing the output, because you can only double so many times. And with an exponentially-growing input, you can get a modest, linear improvement to output. ↩︎

2. i.e. computing for longer or computing more parallel. Parallel can't be better than serial in returns to total compute, so I'm mainly interested in the more generous serial case. For parallel, it's easier to bound because the algorithm space is more constrained ('sample many in parallel, choose best' is the best you can do asymptotically). ↩︎

3. Intuitively you can 'reason deeper' with extra serial compute, which might look like recursing further down a search tree. You can also take proposals and try to refine or improve rather than just throwing them out and trying again from scratch. ↩︎

4. Proof. Suppose the generator produces proposals with quality $X$. All we assume is that the distribution of $X$ has a moment-generating function (this is not true of all distributions, in particular heavy-tailed distributions may not have a MGF). Denote $k$ individual samples as $X_i$. Note first by Jensen's inequality that:

   $$e^{tE\left[{max}_i{X}_i\right]}\le E{e}^{t\left[{max}_i{X}_i\right]}=E\underset{i}{max}{e}^{t{X}_i}$$

   i.e. the exponential of the expected maximum in question is bounded by the expected maximum of the exponentials. But a max of positive terms is bounded by the sum:

   $$E\underset{i}{max}{e}^{t{X}_i}\le E\sum _i{e}^{t{X}_i}=kE{e}^{tX}$$

   (writing $X$ for a representative single sample.) But that's just $k$ times the moment-generating function (which we assumed exists). So for all positive $t$,

   $$E\left[\underset{i}{max}{X}_i\right]\le \frac{\ln k+\ln MGF\left(t\right)}{t}$$

   So (fixing any $t$, or minimising over $t$,) we see at most logarithmic growth in $k$. ↩︎

5. Take the proof of the general case for an arbitrary distribution with a moment-generating function. Substitute the normal moment-generating function $MGF\left(t\right)=e^{\frac{{\sigma }^2{t}^2}{2}}$

   $$E\left[\underset{i}{max}{X}_i\right]\le \frac{\ln k}{t}+\frac{{\sigma }^{2}t}{2}$$

   Minimising over (positive) $t$,

   $$E\left[\underset{i}{max}{X}_i\right]\le \sigma \sqrt{2\ln k}$$ ↩︎

6. Or more than exponential if the order or configuration matters! ↩︎

Discuss