ICML 2025

PAC Learning with Improvements
Idan Attias 1 2 Avrim Blum 2 Keziah Naggita 2 Donya Saless 2 Dravyansh Sharma 2 3 Matthew Walter 2
Abstract
One of the most basic lower bounds in machine
learning is that in nearly any nontrivial setting, it
takes at least 1/ϵ samples to learn to error ϵ (and
more, if the classifier being learned is complex).
However, suppose that data points are agents who
have the ability to improve by a small amount if
doing so will allow them to receive a (desired)
positive classification. In that case, we may ac-
tually be able to achieve zero error by just being
“close enough”. For example, imagine a hiring test
used to measure an agent’s skill at some job such
that for some threshold θ, agents who score above
θ will be successful and those who score below
θ will not (i.e., learning a threshold on the line).
Suppose also that by putting in effort, agents can
improve their skill level by some small amount r.
In that case, if we learn an approximation ˆθ of θ
such that θ ≤ˆθ ≤θ + r and use it for hiring, we
can actually achieve error zero, in the sense that
(a) any agent classified as positive is truly quali-
fied, and (b) any agent who truly is qualified can
be classified as positive by putting in effort. Thus,
the ability for agents to improve has the potential
to allow for a goal one could not hope to achieve
in standard models, namely zero error. In this
paper, we explore this phenomenon more broadly,
giving general results and examining under what
conditions the ability of agents to improve can
allow for a reduction in the sample complexity
of learning, or alternatively, can make learning
harder. We also examine both theoretically and
empirically what kinds of improvement-aware al-
gorithms can take into account agents who have
the ability to improve to a limited extent when it
is in their interest to do so.
Authors are listed alphabetically.
1University of Illinois at
Chicago 2Toyota Technological Institute at Chicago 3Northwestern
University...