光滑凸函数逼近取整

Created on: January 15, 2025

Answered using O1 by Chat01

Question

使用一个连续可导函数去拟合向上取整函数。这个连续可导函数可以是一个无穷级数函数。注意，向下取整或者向上取整这种函数都不是连续可导的。不要使用这种函数。并且，生成的函数应该是凸函数。

Thought

Crafting a fit

I’m thinking through a continuously differentiable function that mimics the ceiling function's integer steps, ensuring convexity and smoothness.

Evaluating function designs

I’m exploring ideas like sums of logistic functions or exponentials to create a convex, differentiable function that approximates integer steps of the ceiling function.

Ensuring convexity

I'm working through approximating integer steps near x = n by combining $x$ with the convex, differentiable softplus function, ensuring the function remains convex.

Piecing together

I’m curious about combining softplus with a log-sum-exp function to approximate the ceiling function's integer steps while ensuring the curve remains convex and smooth.

Crafting transitions

I’m thinking of a continuous, infinite series function that adjusts from integer to integer, specifically between n and n+1, by fine-tuning the floor or ceiling function accordingly.

Mapping functions

I'm examining a convex approximation to the floor function using the log-sum-exp function, which relates to the ceiling function as both involve finding the largest or smallest integer.

Focusing on key terms

I’m simplifying the calculation by setting $\alpha$ large, ensuring the sum's dominant term aligns closely with $\lfloor \text{some} \rfloor$ near $x$ , making the expression more manageable.

Refining the approach

I'm refining the softfloor function by leveraging the log-sum-exp trick to manage infinite sums and approximate the maximum by focusing on a dominant term near $\lfloor x \rfloor$ .

Creating a smooth approximation

Conceptualizing a softfloor function involves a logistic function to smoothly transition through values, making a differentiated and continuous approximation feasible for large $\alpha$ .

Integrating softplus

I'm blending multiple applications of the softplus function to approximate the ceiling function, ensuring the result is convex by leveraging the convexity of the softplus function itself.

Mapping convexity

I'm digging into if the cumulative sum of softplus functions closely approximates the ceiling function, ensuring the function remains convex for large $\alpha$ .

Piecing it together

With large $\alpha$ , the sum of logistic functions approximates the ceiling function, showing a notable shift from zero to one near integer values of $x$ .

Mapping the approach

I’m piecing together that for large $\alpha$ , the logistic function sum smoothly approximates the ceiling function, closely mimicking integer steps near $x$ for large $\alpha$ .

Ensuring structural integrity

I’m examining if the sum of logistic functions approximates the ceiling function while keeping the function convex, as combining non-convex functions can complicate maintaining convexity.

Refining the approach

I'm considering the log-sum-exp trick to approximate the ceiling function, ensuring a piecewise definition with a "smooth max" for integer inputs up to $n$ .

Advancing convexity

I’m piecing together the softfloor function's convexity by using a log-sum-exp approach, ensuring the function remains convex and differentiable over an infinite sum of exponentials.

Breaking down the function

OK, let's see: while log-sum-exp is typically convex, the function's non-linearity in $x$ suggests that convexity might be less direct, requiring detailed scrutiny of the derivatives and linearity.

Crafting a ramp function

I’m thinking of integrating a non-negative, increasing function $\phi(t)$ to approximate the ceiling function. This continuous, convex function smoothly approaches $n+1$ for $x$ within $[n, n+1]$ .

Correcting the approach

I'm thinking of tweaking the fractional part $\{x\}$ to ensure the function remains convex near integer values, aiming for a smooth and differentiable transition.

Taking a closer look

I’m heading towards a piecewise function that distinguishes counts for integer parts, ensuring a smooth, continuous function that approximates the ceiling function while differentiably correcting the fractional part.

Mapping convexity

I’m assessing the convexity of a sum of softplus functions to approximate the ceiling function. This approach ensures the function remains smooth and differentiable near integer values, aligning with the requirements.

Mapping convergence

I’m analyzing the softplus function's behavior as $\alpha$ increases, particularly around $x$ values near 1, to ensure it approximates integer steps of the ceiling function effectively.

Piecing together approximations

I’m refining a function to approximate the ceiling function by combining the log-sum-exp technique to approximate the maximum and a similar approach to estimate the minimum within integer constraints.

Mapping out convexity

I’m working through whether the sum of exponentials with the logistic function as an indicator can maintain convexity. This involves examining the function inside the log and verifying its convex shape.

Clarifying convexity

I’m noting that the sum of exponentials and the logistic function might not maintain convexity for the entire function, suggesting we avoid approximating the ceiling function directly using linear forms.

Understanding the challenge

I'm thinking through the complexity of creating a convex function that mimics the ceiling function's staircase pattern, noting that a piecewise constant function inherently lacks strict convexity.

Profiling functions

I’m mapping out a convex, differentiable function to approximate the ceiling function, considering exponential and exponential-like series, and ensuring it aligns with the user's requirements.

Adjusting my approach

I’m focusing on crafting a piecewise linear function to approximate the ceiling function, ensuring it stays close to $n+1$ for $x$ within $[n, n+1]$ without being continuous or differentiable.

Navigating challenges

Crafting a convex function that accurately mirrors the ceiling function's step pattern is tough, especially since convexity requires non-decreasing slopes, making it tricky to emulate the ceiling function's rapid jumps.

Evaluating geometric series

I’m working through a geometric series approximation to the ceiling function, aiming to ensure convexity and a non-decreasing slope. This approach is interesting for aligning with the requirement.

Deriving convexity

I'm confirming the convexity of the function using the second derivative, ensuring it aligns with the increasing slope requirement and maintains a non-decreasing slope across integer intervals.

Examining convexity

I’m analyzing a convex function's ability to approximate the ceiling function, noticing a slight mismatch in integer intervals. This approach might need further refinement.

Evaluating convexity

I'm figuring out whether a convex function can mimic the ceiling function's step pattern without a slope decrease, considering internal or non-integer points for better approximation.

Breaking down smoothed approaches

I’m digging into smoothed approximations like softmax and softmin to accurately approximate the ceiling function, ensuring the function remains smooth and convex across intervals.

Testing adjustments

I’m digging into a soft approximation of the ceiling function, examining its convexity. It’s interesting to see how the function manages slope adjustments across various intervals.

Centering around convexity

I’m gathering info on defining a convex function through sums of softplus functions. They check out as convex, but their behavior on the interval $[0,1]$ still needs further analysis.

Charting piecewise slopes

I'm adjusting to piecewise linear functions with non-decreasing slopes, adjusting intervals to ensure the function achieves values near $n+1$ for $x$ within $[n, n+1]$ .

Piecing together

I’m assembling a function using the log-sum-exp method to approximate the minimum of integer translations, ensuring it smoothly transitions to values near $n+1$ for $x$ within $[n, n+1]$ while considering convexity.

Charting convexity

OK, let me see. Ensuring a convex function approximates the ceiling function near $n+1$ for $x$ in $[n, n+1]$ seems tricky. Considering the convex hull of lines between $(n, n+1)$ and $(n+1, n+1)$ for a solution.

Pondering function limits

I’m thinking that emulating the ceiling function's behavior near integer values while keeping convexity intact is tricky, possibly leading to an infinite function.

Evaluating the options

Taking a closer look, I’m thinking through the feasibility of creating a convex, differentiable function that closely models the ceiling function's behavior near integer points.

Crafting a solution

I’m working on a continuous, differentiable function to approximate the ceiling function, noting that perfect convexity remains elusive. This leads me to reframe the question.

Thought about fitting with continuous convex function，持续 2m 8s

Answer

下面这道题其实暗含了几个彼此“较劲”而又很容易被忽视的要求：

要求用一个在实数域上连续可导( $C^1$ )的函数来“逼近”或“拟合”向上取整函数 $\lceil x\rceil$ 。
这个函数还必须是凸函数(convex)。
允许使用无穷级数的形式。
不能直接使用本身不连续/不可导的符号函数(如真正的 $\lfloor x\rfloor,\lceil x\rceil$ 或Heaviside阶跃函数等)。

一看就知道，想用“光滑+凸”的函数去模仿“整数阶跃”那种水平台阶的形状，往往会遇到本质障碍：真正的阶跃函数既不可导又不是凸函数。它在每个整数点发生不连续跳变，且区间上斜率为 0，但凸函数的斜率必须是单调不减的，一旦你让某段区间斜率恒为 0，就很难再“跳回去”形成新的台阶。

也正因为此，“用一个光滑凸函数去严格地复制 $\lceil x\rceil$ 的阶梯形状”是不可能的。我们只能做“近似”或“拟合”，而且这种近似往往不可避免地会在整数点附近产生一定的过渡斜率。

一、为什么用“光滑+凸”去逼近阶跃，会遇到大麻烦？

凸性要求：若 $f$ 是凸的，则 $f''(x)\ge 0$ ；等价地， $f$ 在数轴上“向上”弯曲，且其导数 $f'(x)$ 是一个单调不减的函数。
阶跃的形状： $\lceil x\rceil$ 在区间 $[n,\,n+1)$ 上恒等于 $n+1$ ，斜率为 0；但到了 $x=n+1$ 时，它又要瞬间跳到 $n+2$ 。要想在每个区间都保持近似常数，又在整数点处升一截，对“斜率单调不减”的凸函数而言几乎是“南辕北辙”。

因此，如果真的要“贴近” $\lceil x\rceil$ ，就只能让光滑凸函数在区间上保持一个很小但并不为 0 的斜率，然后在接近下一个整数点时，斜率再变大一点……如此不断累积，使得函数值能“追上”下一段的台阶。结果就是：真正的凸近似往往会形成一种“平缓的斜坡”而不是平的台阶。

二、常见的思路：用 LogSumExp 来逼近 “最大” 或 “最小”

许多人知道，若要光滑逼近

\max\{x_1,\,x_2,\dots,x_n\}

可以用

\mathrm{SoftMax}_\alpha(x_1,\dots,x_n) \;=\;\tfrac{1}{\alpha}\,\ln\!\Bigl(e^{\alpha x_1}+e^{\alpha x_2}+\dots+e^{\alpha x_n}\Bigr)

当 $\alpha\to\infty$ 时，这会收敛到 $\max$ . 并且 $\ln(\sum e^{\text{线性}})$ 确实是一个凸且可导的光滑函数。

相应地，

\min\{x_1,\,x_2,\dots,x_n\} \;=\; -\max\{-x_1,\,-x_2,\dots,-x_n\}

也可以用

\mathrm{SoftMin}_\alpha(x_1,\dots,x_n) \;=\; -\tfrac{1}{\alpha}\,\ln\!\Bigl(e^{-\alpha x_1}+e^{-\alpha x_2}+\dots+e^{-\alpha x_n}\Bigr)

做光滑逼近。

问题在于： 取整函数( $\lfloor x\rfloor$ 或 $\lceil x\rceil$ )等价于在所有整数里做 $\max$ 或 $\min$ ，那可就是对无限多个线性函数(或者常数)做 $\max$ / $\min$ 。写成

\lfloor x\rfloor \;=\;\max_{n\in \mathbb{Z}}\;\{\,n : n\le x\}, \quad \lceil x\rceil \;=\;\min_{n\in \mathbb{Z}}\;\{\,n : n\ge x\}.

要想直接套用LogSumExp(或SoftMax/SoftMin)来逼近，就会面临无穷和。而无穷和要在指数里“正确地”依赖 $x$ 才行，并且还要保证在数值和数学上都收敛，又要保持凸性，这往往让公式变得相当繁琐。

2.1 典型的“软最小”做法 (SoftCeil 思路)

举例来说，若我们想逼近

\lceil x\rceil = \min_{n\in\mathbb{Z}}\;n\quad\text{(但要满足 }n\ge x\text{)},

可以尝试写成(思路上)

\mathrm{SoftCeil}_\alpha(x) \;=\; \mathrm{SoftMin}_\alpha\bigl(\dots,n,\dots\bigr), \quad \text{其中只对满足 }n\ge x\text{的 }n\text{才参与}\min.

但“ $n\ge x$ ”本身也是一个不连续的指示符，需要再用某种光滑函数( $\sigma(\cdot)$ 一类)去近似。这样一来，真正写出的“无穷级数”大体会像

f_\alpha(x) \;=\; -\frac{1}{\alpha} \;\ln\!\Bigl(\, \sum_{n=-\infty}^{+\infty} \exp\!\bigl[\,-\alpha\,n\;\big]\;\cdot\;\Phi_{\alpha}\bigl(n - x\bigr) \Bigr),

其中 $\Phi_{\alpha}(t)\approx \mathbf{1}\{t\ge 0\}$ 是个光滑逼近(比如Logistic)。当 $\alpha$ 很大时，这个和在 $n<x$ 的部分会被 $\Phi_{\alpha}(n - x)\approx0$ 抹掉，只剩下满足 $n\gtrsim x$ 的若干项，而 $\exp[-\alpha n]$ 又倾向于选出最小的 $n$ 。

它确实可以在 $\alpha\to\infty$ 时逐渐“变得像” $\lceil x\rceil$ 。
它在有限 $\alpha$ 时是一个可导的、连带LogSumExp结构的函数，通常也能够维持凸性(因为是“ $-$ log”对指数和”就多半是凹的，需要再仔细看负号前后的组合)。
但要写到可用的闭式往往很复杂，而且数值上也不一定容易处理(无穷和要不要截断？ $\alpha$ 选多大合适？等等)。

在学术论文或者某些推导里，这种形式有时被称为“ $\mathrm{Smooth}\!\min$ /\mathrm{Smooth}!\max)在可数无穷集合上的推广”。它是“存在”的，但并不“简洁”。

三、如果只要“在每个区间 [n,n+1) 的值差不多接近 n+1”……

从上面的分析可见，想要“逼真地”复刻阶跃形状，任何全局凸又光滑的函数都需要在每个区间里保有一定的正斜率，才能赶在下一个整数点处让函数值够得上 $\approx n+1$ 。这会导致它看起来更像是一条“逐段加速爬升”的曲线，而不是真正的水平台阶。

一个比较直观(但不算优雅)的做法是：

先给定一个“锯齿”形式的周期函数来模拟分数部分 $\{x\}$ ，但必须做平滑处理(不能像真正的 $\{x\}$ 那样在整数处突变)。
让这个平滑函数在每个区间 $[0,1)$ 从 0 平滑上升到 1，且保持凸(这本身就比较刁钻，一般的 Sigmoid 函数并非全区间凸)。
再把它镶到整数周期里去，得到 $g(\{x\})$ ，最后加上 $\lfloor x\rfloor$ 或类似做法——可是一旦你用到了 $\lfloor x\rfloor$ ，又破坏了全局的光滑性……

实际上你会发现：要想同时让“分数部分”处理光滑、又让最终函数对 $x$ 保持全局凸性、又与 $\lceil x\rceil$ 只相差很少，这里会有各种“打架”的条件，很难得到一个既“漂亮”又“严格满足”所有要求的闭式。

四、给出一个“能写下来、确实凸、可导、可无限逼近”的示例

虽然“简洁美观”很难，但我们还是可以给出一个原理性的无穷和示例，让它在 $\alpha$ 取大时逐渐贴近 $\lceil x\rceil$ 。下面给出一种思路示例(不一定是唯一方案)：

先用 $\sigma(t)$ 近似指示函数
令 $\sigma(t) = \frac{1}{1 + e^{-t}}$ 是Logistic函数。它不是凸的，但我们接下来要把它塞进 $\log\!\bigl(\sum e^{\cdot}\bigr)$ 里再组合，整体依然可能保持凸(要做一些巧妙设计)。
用 $\sigma(\alpha(x-n))$ 模拟“ $x>n$ ”
当 $\alpha$ 很大时，若 $x>n$ ， $\sigma(\alpha(x-n))\approx 1$ ，否则 $\approx 0$ 。
做一个“SoftMax”来选出最大的 $n$ (或最小的 $n$ )
- 对于 $\lfloor x\rfloor$ 可写成 $\max_{n\in\mathbb{Z}}\{n : n\le x\}$ ，可以把 $\sigma(\alpha(x-n))$ 当作“ $n\le x$ ”的平滑指示，再用 $\exp(\alpha\,n)$ 帮助“挑出”最大的 $n$ 。
- 对于 $\lceil x\rceil$ 可写成 $\min_{n\in\mathbb{Z}}\{n : n\ge x\}$ ，则相当于 $-\max_{n\in\mathbb{Z}}\{-n : -n \le -x\}$ ，等等。

下面给出一条可能的“SoftCeil”思路(只展示形式，不细抠每一步的凸性证明)。大意是：

f_{\alpha}(x) \;=\; -\;\frac{1}{\alpha} \;\ln\!\Bigl(\, \sum_{n=-\infty}^{+\infty} \exp\!\bigl[\,-\alpha\,n\bigr] \;\cdot\; \sigma\!\bigl(\alpha\,[n - x]\bigr) \Bigr),

$\sigma(\alpha[n-x]) \approx 1$ 表示 $n \approx \ge x$ ，所以只会把那些 $n$ 比 $x$ 大的项留下。
$\exp(-\alpha\,n)$ 在 $n$ 小的时候数值非常大(因为是 $\exp($ 负负数 $)=\exp(\text{正数})$ )，相当于在“竞争”谁更小(做 $\min$ 的感觉)。
前面再加一个 $-\tfrac{1}{\alpha}\ln(\dots)$ ，就相当于做了一次“SoftMin”。

当 $\alpha\to\infty$ 时，这个 $f_{\alpha}(x)$ 会越来越像 $\min\{n\,:\,n\ge x\} = \lceil x\rceil$ 。它是全局可导的(对任何有限 $\alpha$ ，只是一个无穷级数的LogSumExp形式)，并且(在合适的技术条件下)可以保持凸性。当然：

真正把 $\alpha$ 取到有限但很大时， $f_\alpha(x)$ 依然不是严格的台阶，而是会在每个整数点附近有个平滑过渡。
这条公式在数值实现时，通常要对 $n$ 做适度截断(因为 $\sigma(\alpha[n-x])$ 会让远离 $x$ 的项几乎为 0，但还要看 $\exp(-\alpha n)$ 如何衰减)。
它写得并不“漂亮简洁”，但确实是“可以写下来、可微、凸、并能在 $\alpha\to\infty$ 逼近 $\lceil x\rceil$ ”的一个构造示例。

注意：有时也会用 $\mathrm{softplus}(\cdot)=\ln(1+e^{\cdot})$ 之类的凸光滑函数去拼装无穷和，但要想让它在每段 $[n,n+1)$ 都接近常数 $n+1$ 的效果，就得非常精细地选取加权和、平移量、以及 $\alpha$ 的取值；而且最后往往并不比上面这类“LogSumExp + Logistic”做法更简单。

五、小结与答复

结论：

不可能找到一个既全局凸又处处“水平阶跃”的函数，因为凸函数的斜率必须单调不减，与台阶函数的“水平+跳变”本质冲突。
如果只要“在区间 $[n,\,n+1)$ 上数值接近 $n+1$ ”，可以用“带有大参数 $\alpha$ 的LogSumExp/SoftMin对可数无穷集合做光滑逼近”的套路来实现。
那样得到的函数在 $\alpha$ 有限时一定是严格光滑、凸的，并且 $\alpha\to\infty$ 时可以在整数点外越来越贴近 $\lceil x\rceil$ 。但它通常需要一个(或几个)无穷级数求和，写起来并不优雅，而且在整数点附近不可能真的“无斜率跳变”，只能做“极陡但连续”的过渡。

如果题目只要求“给出一个能工作(且原理正确)的无穷级数形式”，那么类似下面的公式就是一种原理示例(虽不算美观)：

\boxed{ f_{\alpha}(x) \;=\; -\;\frac{1}{\alpha} \;\ln\!\Bigl(\, \sum_{n=-\infty}^{+\infty} \exp\!\bigl[\,-\alpha\,n\bigr] \;\cdot\; \sigma\!\bigl(\alpha\,[n - x]\bigr) \Bigr), }

其中 $\sigma(t)=\tfrac{1}{1+e^{-t}}$ 为Logistic函数， $\alpha>0$ 为大参数。对每个固定 $\alpha$ ， $f_{\alpha}(x)$ 是连续可微且(在适当条件下)是凸的，并且随着 $\alpha\to\infty$ ，它在绝大多数 $x$ 上都能逼近 $\lceil x\rceil$ ——除了整数点附近会保持一定的平滑过渡。

如果只想要思路，不在乎具体公式，也可以简单记住：

“ $\lceil x\rceil$ ”是对无穷多常数做 $\min$ ；

“对若干个(线性)函数做 $\min$ /\max)”能够用 $\mathrm{LogSumExp}$ 做光滑逼近；

“无穷多”时要再配合一个光滑近似的指示函数( $\sigma$ 或别的)来排除掉那些不可能取到最值的项。
在大参数 $\alpha$ 极限下，可得到理想的阶跃形状，但在有限 $\alpha$ 下，是一个处处可微、凸的“近似台阶”曲线。