Previously we saw that K-means stuck at local optimal if seed = 100:

Without changing the seed and K, add only one training example to save K-means out of the local optimal situation (ie achieving a low training error). Write down this additional training example below.

https://stanford-cs221.github.io/autumn2019/lectures/index.html#include=learning-demo.js&example=cluster

Question

Previously we saw that K-means stuck at local optimal if seed = 100:

Without changing the seed and K, add only one training example to save K-means out of the local optimal situation (ie achieving a low training error). Write down this additional training example below.

https://stanford-cs221.github.io/autumn2019/lectures/index.html#include=learning-demo.js&example=cluster

BlackTom AI · Accepted Answer

The prompt presents a short-answer task, not a multiple-choice question, so there are no answer options to evaluate. Instead, here is a structured way to reason about adding a single training example to help K-means escape a local optimum when the seed and K are fixed.

- Understand the problem setting: With a fixed seed, K-means initialization is deterministic, so the algorithm may converge to a local minimum that depends on the initial centers and the data distribution. By adding one carefully chosen point, you can nudge the centroid updates in a way that leads to a different, lower-final training error after re-running K-means.

- Conceptual goal: The added point should influence the movement of at least one cluster center so that the subsequent assignments/re-computations reduce the overall within-cluster sum of squares (WCSS). In practice, you want a point that pulls a center toward a region that better represents a true cluster, or that breaks an unfavorable symmetry that caused the local optimum.

- Strategy 1: Add a point near an existing cluster center but slightly offset toward the boundary between two clusters. This can cause the affected center to shift toward a more representative location, especially if the data originally formed a tight, overlapping configuration where a minor perturbation changes centroid placements.
  - Why it might work: The added point acts as a soft attractor/bias, altering the mean of its assigned cluster, which can cascade into more favorable reassignments in subsequent iterations.
  - How to think about placement: Place the point within the vicinity of a cluster but not exactly at its center; a small offset toward the neighboring cluster helps the algorithm reallocate points more evenly if there is ambiguity near the boundary.

- Strategy 2: Add a point that is clearly closer to a different cluster than its surrounding points would imply under the current centers. This can trigger a reallocation that reduces within-cluster distances for both affected clusters.
  - Why it might work: It creates a counterexample to the current assignment, encouraging a re-partitioning that yields a lower WCSS after convergence.
  - How to think about placement: Choose a location that is decisively closer to one cluster center but benefits the global configuration if that center shifts a bit toward it.

- Strategy 3: Place a point that reinforces a genuine separate cluster that is underrepresented or barely detected in the current run. If the single point forms a small but coherent sub-group with existing points, the updated centers may better capture the true structure.
  - Why it might work: It increases the weight of a small cluster in the centroid calculation, potentially leading to a more accurate partitioning overall.
  - How to think about placement: If you observe a suspected third cluster in the data, insert a point that belongs clearly to that cluster’s region without duplicating existing points.

- Practical considerations:
  - Keep the seed and K unchanged, as specified, and focus on data placement rather than initialization.
  - The exact numeric coordinates depend on the original dataset geometry; the key idea is to introduce a perturbation that alters centroid movement during the first or second iteration.
  - After adding the point, re-run K-means and compare the final WCSS to the previous run to verify improvement.

- If you must provide a concrete example: choose a coordinate that lies near the boundary between two clusters or within a low-density area that, when included, shifts a centroid toward a more representative center. Ensure the point is not so far from all existing points that it creates an entirely new, dominant outlier cluster unless the data actually warrants it.

In short, the added example should act as a targeted perturbation: it nudges one or more centroids in a direction that aligns better with the underlying data structure, thereby enabling K-means to converge to a lower training error under the fixed seed and K.

Artificial Intelligence Lecture 4 quiz

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