I would like to understand intuitively what it means to sample from a distribution $\mathcal{D}$. It may sound like a dumb question, but I can't find an answer anywhere, a colleague recommended sharing the question with this cs community as there are many brilliant researchers with nice intuitions.

In the context of active learning, we sample from regions of the input space where the probability mass is high and hope to learn the model on those regions. How does it happen in practice when we don't know the underlying distribution $\mathcal{D}$ (stream-based active learning), but only have access to a finite dataset sampled from the distribution (pool-based active learning)? Is streaming-based active learning not feasible in practice?

In some papers (e.g, this paper) authors write $x \sim \mathcal{D}$ to say we use iid sampling from the distribution. In the sequential setting, when sampling $x_t$ at time step $t$ and $x_t'$ at time step $t'$ such that $t < t'$, does it mean $\underset{x \sim \mathcal{D}}{\mathbb{P}}[x = x_t] > \underset{x \sim \mathcal{D}}{\mathbb{P}}[x = x_{t'}]$? In other words, in the sequential setting, do we guarantee that we always sample from high-probability regions at first?

I would be grateful to receive any insights.


1 Answer 1


Active learning involves choosing which point to query for a label -- as opposed to passive learning, where the example-label pairs are drawn from a distribution without any control of the learner.

See the paper Active Nearest-Neighbor Learning in Metric Spaces Aryeh Kontorovich, Sivan Sabato, Ruth Urner; JMLR 18(195):1−38, 2018. https://www.jmlr.org/papers/v18/16-499.html

for a detailed discussion

  • $\begingroup$ Thank you @Aryeh, I understand that the distribution controls learning in the passive setting but that's only for the pool-based active learning. However, as I understand in streaming-based active learning, the model should perform well in high-probability mass regions. If the queries do not take care of those regions (distribution is unknown), how can we ensure that the model won't make a mistake on a freshly sampled point from the distribution? $\endgroup$ Commented May 7 at 12:03
  • $\begingroup$ can you link to a paper where this model is rigorously defined? $\endgroup$
    – Aryeh
    Commented May 7 at 16:07
  • 1
    $\begingroup$ Paper jmlr.org/papers/v13/dekel12b.html: Selective Sampling and Active Learning from Single and Multiple Teachers by Ofer Dekel, Claudio Gentile, Karthik Sridharan. This Paper: proceedings.neurips.cc/paper/2007/file/… is more classic in active learning and describes my confusion, where the setting is explained rigorously in section 2.1, and my confusion appears in the first line of Algorithm 1 (Input) : A general agnostic active learning algorithm by Sanjoy Dasgupta, Daniel Hsu and Claire Monteleoni $\endgroup$ Commented May 7 at 16:39

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