A general agnostic active learning algorithm
Sanjoy Dasgupta, Daniel Hsu, and Claire Monteleoni
University of California, San Diego
{dasgupta,djhsu,cmontel}@cs.ucsd.edu
Abstract
We present a simple, agnostic active learning algorithm
that works for any hypothesis class of bounded VC dimension,
and any data distribution. Our algorithm extends
a scheme of Cohn, Atlas, and Ladner to the agnostic
setting, by (1) reformulating it using a reduction
to supervised learning and (2) showing how to apply
generalization bounds even for the non-i.i.d. samples
that result from selective sampling. We provide a
general characterization of the label complexity of our
algorithm. This quantity is never more than the usual
PAC sample complexity of supervised learning, and is
exponentially smaller for some hypothesis classes and
distributions. We also demonstrate improvements experimentally.
Introduction
Active learning addresses the issue that, in many applications,
labeled data typically comes at a higher cost (e.g. in
time, effort) than unlabeled data. An active learner is given
unlabeled data and must pay to view any label. The hope
is that significantly fewer labeled examples are used than in
the supervised (non-active) learning model. Active learning
applies to a range of data-rich problems such as genomic
sequence annotation and speech recognition. In this paper
we formalize, extend, and provide label complexity guarantees
for one of the earliest and simplest approaches to active
learning—one due to Cohn, Atlas, and Ladner (1994).
The scheme of Cohn, Atlas, and Ladner examines data one
by one in a stream and requests the label of any data point
about which it is currently unsure. For example, suppose the
hypothesis class consists of linear separators in the plane,
and assume that the data is linearly separable. Let the first
six data be labeled as follows.
01
01 01
01 01
01
Copyright c○ 2007, Association for the Advancement of Artificial
Intelligence (www.aaai.org). All rights reserved.
The learner does not need to request the label of the seventh
point (indicated by the arrow) because it is not unsure about
the label: any straight line with the ⊕s and ⊖s on opposite
sides has the seventh point with the ⊖s. Put another way,
the point is not in the region of uncertainty (Cohn, Atlas,
and Ladner, 1994), the portion of the data space for which
there is disagreement among hypotheses consistent with the
present labeled data.
Although very elegant and intuitive, this approach to active
learning faces two problems:
1. Explicitly maintaining the region of uncertainty can be
computationally cumbersome.
2. Data is usually not perfectly separable.
Our main contribution is to address these problems. We
provide a simple generalization of the selective sampling
scheme of Cohn, Atlas, and Ladner that tolerates adversarial
noise and never requests many more labels than a standard
agnostic supervised learner would to learn a hypothesis with
the same error.
In the previous example, an agnostic active learner (one that
does not assume a perfect separator exists) is actually still
uncertain about the label of the seventh point, because all
six of the previous labels could be inconsistent with the best
separator. Therefore, it should still request the label. On
the other hand, after enough points have been labeled, if an
unlabeled point occurs at the position shown below, chances
are its label is not needed.
01 01 01 01 01
01 01 0100 1101
01 01
01
01 01 01
01
01
01
01 01
01 01 01
01 01 01 01
To extend the notion of uncertainty to the agnostic setting,
we divide the observed data points into two groups, ˆ S and
T :
• Set ˆ S contains the data for which we did not request labels.
We keep these points around and assign them the
label we think they should have.
• Set T contains the data for which we explicitly requested
labels.We will manage things in such a way that the data in ˆ S are
always consistent with the best separator in the class. Thus,
somewhat counter-intuitively, the labels in ˆ S are completely
reliable whereas the labels in T could be inconsistent with
the best separator. To decide whether we are uncertain about
the label of a new point x, we reduce to supervised learning:
we learn hypotheses h+1 and h−1 such that
• h+1 is consistent with all the labels in ˆ S ∪ {(x,+1)} and
has minimal empirical error on T , while
• h−1 is consistent with all the labels in ˆ S ∪ {(x, −1)} and
has minimal empirical error on T .
If, say, the true error of the hypothesis h+1 is much larger
than that of h−1, we can safely infer that the best separator
must also label x with −1 without requesting a label; if the
error difference is only modest, we explicitly request a label.
Standard generalization bounds for an i.i.d. sample let us
perform this test by comparing empirical errors on ˆ S ∪ T .
The last claim may sound awfully suspicious, because ˆ S ∪T
is not i.i.d.! Indeed, this is in a sense the core sampling
problem that has always plagued active learning: the labeled
sample T might not be i.i.d. (due to the filtering of examples
based on an adaptive criterion), while ˆ S only contains unlabeled
examples (with made-up labels). Nevertheless, we
prove that in our case, it is in fact correct to effectively pretend
ˆ S ∪ T is an i.i.d. sample. A direct consequence is that
the label complexity of our algorithm (the number of labels
requested before achieving a desired error) is never much
more than the usual sample complexity of supervised learning
(and in some cases, is significantly less).
An important algorithmic detail is the specific choice of generalization
bound we use in deciding whether to request a
label or not. A small polynomial difference in generalization
rates (between n −1/2 and n −1 , say) can get magnified
into an exponential difference in label complexity, so it is
crucial for us to use a good bound. We use a normalized
bound that takes into account the empirical error (computed
on ˆ S ∪ T —again, not an i.i.d. sample) of the hypothesis in
question.
Earlier work on agnostic active learning (Balcan, Beygelzimer,
and Langford, 2006; Hanneke, 2007) has been able to
upper bound label complexity in terms of a parameter of the
hypothesis class (and data distribution) called the disagreement
coefficient. We give label complexity bounds for our
method based on this same quantity, and we get a better dependence
on it, linear rather than quadratic.
To summarize, in this paper we present and analyze a simple
agnostic active learning algorithm for general hypothesis
classes of bounded VC dimension. It extends the selective
sampling scheme of Cohn, Atlas, and Ladner (1994) to the
agnostic setting, using normalized generalization bounds,
which we apply in a simple but subtle manner. For certain
hypothesis classes and distributions, our analysis yields improved
label complexity guarantees over the standard sample
complexity of supervised learning. We also demonstrate
such improvements experimentally.
Related work
A large number of algorithms have been proposed for active
learning, under a variety of learning models. In this
section, we consider only methods whose generalization behavior
has been rigorously analyzed.
An early landmark result was the selective sampling scheme
of Cohn, Atlas, and Ladner (1994) described above. This
simple active learning algorithm, designed for separable
data, has been the inspiration for a lot of subsequent work. A
few years later, the seminal work of Freund, Seung, Shamir,
and Tishby (1997) analyzed an algorithm called query-bycommittee
that operates in a Bayesian setting and uses an
elegant sampling trick for deciding when to query points.
The core primitive required by this algorithm is the ability
to sample randomly from the posterior over the hypothesis
space. In some cases this can be achieved efficiently (Gilad-
Bachrach, Navot, and Tishby, 2005), for instance when the
hypothesis class consists of linear separators in R d (with a
uniform prior) and the data is distributed uniformly over the
surface of the unit sphere in R d . In this particular setting,
the authors showed that the number of labels required to
achieve generalization error ε is just O(d log 1/ε), exponentially
lower than the usual supervised sample complexity of
O(d/ε).
Subsequently, Dasgupta, Kalai, and Monteleoni (2005)
showed that a simple variant of the perceptron algorithm
also achieves this label complexity, even for a worst-case
(non-Bayesian) choice of target hypothesis.
All the work mentioned so far assumes separable data. This
case was studied abstractly by Dasgupta (2005), who found
that a parameter called the splitting index loosely characterizes
the label complexity of actively learning hypothesis
classes of bounded VC dimension. As yet, it is not known
how to realize this label complexity in a computationally efficient
way, except in special cases.
A natural way to formulate active learning in the agnostic
setting is to ask the learner to return a hypothesis with error
at most ν + ε (where ν is the error of the best hypothesis
in the specified class) using as few labels as possible. A
basic constraint on the label complexity was pointed out by
Kääriäinen (2006), who showed that for any ν ∈ (0,1/2),
there are data distributions that force any active learner that
achieves error at most ν + ε to request Ω((ν/ε) 2 ) labels.
The first rigorously-analyzed agnostic active learning algorithm,
called A2 , was developed recently by Balcan,
Beygelzimer, and Langford (2006). Like Cohn-Atlas-
Ladner (1994), this algorithm uses a region of uncertainty,
although the lack of separability complicates matters and
A 2 ends up explicitly maintaining an ε-net of the hypothesis
space. Subsequently, Hanneke (2007) characterized the
label complexity of the A 2 algorithm in terms of a parameter
called the disagreement coefficient. Another thread of
work focuses on agnostic learning of thresholds for data thatlie on a line; in this case, a precise characterization of label
complexity can be given (Castro and Nowak, 2006, 2007).
These previous results either make strong distributional assumptions
(such as separability, or a uniform input distribution),
or else they are computationally prohibitive in general.
Our work was inspired by both Cohn-Atlas-Ladner and
Balcan-Beygelzimer-Langford, and we have built heavily
upon their insights. We bound the label complexity of our
method in terms of the same parameter as used for A 2 (Hanneke,
2007), and get a somewhat better dependence (linear
rather than quadratic).
A common feature of Cohn-Atlas-Ladner, A 2 , and our
method is that they are all fairly non-aggressive in their
choice of query points. They are content with querying all
points on which there is even a small amount of uncertainty,
rather than, for instance, pursuing the maximally uncertain
point. Recently, Balcan, Broder, and Zhang (2007) showed
that for the hypothesis class of linear separators, under distributional
assumptions on the data (for instance, a uniform
distribution over the unit sphere), a more aggressive strategy
can yield better label complexity.
Preliminaries
Learning framework and uniform convergence
Let X be the input space, D a distribution over X ×{±1} and
H a class of hypotheses h : X → {±1} with VC dimension
vcdim(H) = d < ∞. Recall that the nth shattering coefficient
S(H,n) is defined as the maximum number of ways in
which H can label a set of n points; by Sauer’s lemma, this
is at most O(n d ) (Bousquet, Boucheron, and Lugosi, 2004,
p.175). We denote by DX the marginal of D over X . In our
active learning model, the learner receives unlabeled data
sampled from DX ; for any sampled point x, it can optionally
request the label y sampled from the conditional distribution
at x. This process can be viewed as sampling (x,y) from
D and revealing only x to the learner, keeping the label y
hidden unless the learner explicitly requests it. The error of
a hypothesis h under D is errD(h) = Pr (x,y)∼D[h(x) ̸= y],
and on a finite sample Z ⊂ X × {±1}, the empirical error
of h is
err(h,Z) = 1
|Z|
∑
(x,y)∈Z
1l[h(x) ̸= y],
where 1l[·] is the 0-1 indicator function. We assume for simplicity
that the minimal error ν = inf{errD(h) : h ∈ H} is
achieved by a hypothesis h ∗ ∈ H.
Our algorithm and analysis use the following normalized
uniform convergence bound (Bousquet, Boucheron, and Lugosi,
2004, p.200).
Lemma 1 (Vapnik and Chervonenkis (1971)). Let F be a
family of measurable functions f : Z → {0,1} over a space
Z. Denote by EZf the empirical average of f over a subset
Z ⊂ Z. Let αn = √ (4/n)ln(8S(F,2n)/δ). If Z is an
i.i.d. sample of size n from a fixed distribution over Z, then,
with probability at least 1 − δ, for all f ∈ F:
( √
−min EZf,α 2 √ )
n + αn Ef
≤
≤
αn
Disagreement coefficient
Ef − EZf
(
min α 2 √ √ )
n + αn EZf,αn Ef .
The active learning algorithm we will shortly describe is not
very aggressive: rather than seeking out points that are maximally
informative, it queries every point that it is somewhat
unsure about. The early work of Cohn-Atlas-Ladner (1994)
and the recent A2 algorithm (Balcan, Beygelzimer, and
Langford, 2006) are similarly mellow in their querying strategy.
The label complexity improvements achievable by
such algorithms are nicely captured by a parameter called
the disagreement coefficient, introduced recently by Hanneke
(2007) in his analysis of A2 .
To motivate the disagreement coefficient, imagine that we
are in the midst of learning, and that our current hypothesis
ht has error at most β. Suppose we even know the value
of β. Then the only candidate hypotheses we still need to
consider are those that differ from ht on at most a 2β fraction
of the input distribution, because all other hypotheses must
have error more than β. To make this a bit more formal, we
impose a (pseudo-)metric ρ on the space of hypotheses, as
follows.
Definition 1. The disagreement pseudo-metric ρ on H is
defined by
ρ(h,h ′ ) = Pr [h(x) ̸= h
x∼DX
′ (x)]
for h,h ′ ∈ H. Let B(h,r) = {h′ ∈ H : ρ(h,h ′) ≤ r} be
the ball centered around h of radius r.
Returning to our earlier scenario, we need only consider hypotheses
in B(ht,2β) and thus, when we see a new data
point x, there is no sense in asking for its label if all of
B(ht,2β) agrees on what this label should be. The only
points we potentially need to query are
{x : h(x) ̸= h ′ (x) for some h,h ′ ∈ B(ht,2β)}.
Intuitively, the disagreement coefficient captures how the
measure of this set grows with β. The following is a slight
variation of the original definition of Hanneke (2007).
Definition 2. The disagreement coefficient θ =
θ(D, H,ε) > 0 is
{
Prx∼DX
inf
r≥ε+ν
[∃h ∈ B(h∗ ,r) s.t. h(x) ̸= h∗ }
(x)]
r
where h ∗ = arg infh∈H errD(h) and ν = errD(h ∗ ).
Clearly, θ ≤ 1/(ε+ν); furthermore, it is a constant bounded
independently of 1/(ε + ν) in several cases previously considered
in the literature (Hanneke, 2007). For example, if
H is homogeneous linear separators and DX is the uniform
distribution over the unit sphere in R d , then θ = Θ( √ d).Algorithm 1
Input: stream (x1,x2,...,xm) i.i.d. from DX
Initially, ˆ S0 := ∅ and T0 := ∅.
For n = 1,2,...,m:
1. For each ˆy ∈ {±1}:
let hˆy := LEARNH( ˆ Sn−1 ∪ {(xn, ˆy)},Tn−1).
2. If err(h−ˆy, ˆ Sn−1 ∪ Tn−1) − err(hˆy, ˆ Sn−1 ∪ Tn−1) >
∆n−1 (or if no such h−ˆy is found) for some ˆy ∈ {±1},
then ˆ Sn := ˆ Sn−1 ∪ {(xn, ˆy)} and Tn := Tn−1.
3. Else request yn; ˆ Sn := ˆ Sn−1 and Tn := Tn−1 ∪
{(xn,yn)}.
Return hf = LEARNH( ˆ Sm,Tm).
Figure 1: The agnostic selective sampling algorithm. See
(1) for a possible setting for ∆n.
Agnostic selective sampling
Here we state and analyze our general algorithm for agnostic
active learning. The main techniques employed by the
algorithm are reductions to a supervised learning task and
generalization bounds applied to differences of empirical errors.
A general algorithm for agnostic active learning
Figure 1 states our algorithm in full generality. The input is
a stream of m unlabeled examples drawn i.i.d from DX ; for
the time being, m can be thought of as Õ((d/ε)(1 + ν/ε))
where ε is the accuracy parameter. 1
The algorithm operates by reduction to a special kind of supervised
learning that includes hard constraints.
For A,B ⊂ X × {±1}, let LEARNH(A,B) denote
a supervised learner that returns a hypothesis h ∈ H
consistent with A, and with minimum error on B. If
there is no hypothesis consistent with A, it reports this.
For some simple hypothesis classes like intervals on the line,
or rectangles in R 2 , it is easy to construct such a learner. For
more complex classes like linear separators, the main bottleneck
is the hardness of minimizing the 0 − 1 loss on B (that
is, the hardness of agnostic supervised learning). If a convex
upper bound on this loss function is used instead, as in the
case of soft-margin support vector machines, it is straightforward
to incorporate hard constraints; but at present the
rigorous guarantees accompanying our algorithm apply only
if 0 − 1 loss is used.
Algorithm 1 maintains two sets of labeled examples, ˆ S and
T , each of which is initially empty. Upon receiving xn, it
learns two 2 hypotheses, hˆy = LEARNH( ˆ S ∪ {(xn, ˆy)},T)
1
The Õ notation suppresses log 1/δ and terms polylogarithmic
in those that appear.
2
If LEARNH cannot find a hypothesis consistent with Sˆ ∪
for ˆy ∈ {±1}, and then compares their empirical errors on
ˆS ∪T . If the difference is large enough, it is possible to infer
how h ∗ labels xn (as we show in Lemma 3). In this case, the
algorithm adds xn, with this inferred label, to ˆ S. Otherwise,
the algorithm requests the label yn and adds (xn,yn) to T .
Thus, ˆ S contains examples with inferred labels consistent
with h ∗ , and T contains examples with their requested labels.
Because h ∗ might err on some examples in T , we just
insist that LEARNH find a hypothesis with minimal error
on T . Meanwhile, by construction, h ∗ is consistent with ˆ S
(as we shall see), so we require LEARNH to only consider
hypotheses consistent with ˆ S.
Bounds for error differences
We still need to specify ∆n, the threshold value for error
differences that determines whether the algorithm requests a
label or not. Intuitively, ∆n should reflect how closely empirical
errors on a sample approximate true errors on the distribution
D. Note that our algorithm is modular with respect
to the choice of ∆n, so, for example, it can be customized
for a particular input distribution and hypothesis class. Below
we provide a simple and adaptive setting that works for
any distribution and hypothesis class with finite VC dimension.
The setting of ∆n can only depend on observable quantities,
so we first clarify the distinction between empirical errors on
ˆSn ∪ Tn and those with respect to the true (hidden) labels.
Definition 3. Let ˆ Sn and Tn be as defined in Algorithm 1.
Let Sn (shedding the hat accent) be the set of labeled examples
identical to those in ˆ Sn, except with the true hidden
labels swapped in. Thus, for example, Sn ∪ Tn is an
i.i.d. sample from D of size n. Finally, let
errn(h) = err(h,Sn∪Tn) and êrrn(h) = err(h, ˆ Sn∪Tn).
It is straightforward to apply Lemma 1 to empirical errors on
Sn ∪ Tn, i.e. to errn(h), but we cannot use such bounds algorithmically:
we do not request the true labels for points in
ˆSn and thus cannot reliably compute errn(h). What we can
compute are error differences errn(h) − errn(h ′ ) for pairs
of hypotheses (h,h ′ ) that agree on (and thus make the same
mistakes on) ˆ Sn, since for such pairs, we have
errn(h) − errn(h ′ ) = êrrn(h) − êrrn(h ′ ). 3
These empirical error differences are means of {−1,0,+1}valued
random variables. We need to rewrite them in terms
of {0,1}-valued random variables for some of the concentration
bounds we will be using.
{(xn, y)} for some y, then assuming h ∗ is consistent with ˆ S, it
must be that h ∗ (x) = −y. In this case, we simply add (xn, −y) to
ˆS, regardless of the error difference.
3
This observation is enough to immediately justify the use of
additive generalization bounds for ∆n. However, we need to use
normalized (multiplicative) bounds to achieve a better label complexity.Definition 4. For a pair (h,h′) ∈ H × H, define
g +
h,h ′(x,y) = 1l[h(x) ̸= y ∧ h ′(x) = y] and g −
h,h ′(x,y) =
1l[h(x) = y ∧ h ′(x) ̸= y].
With this notation, we have err(h,Z) − err(h ′,Z) =
EZ[g +
h,h ′] − EZ[g −
h,h ′] for any Z ⊂ X × {±1}. Now, applying
Lemma 1 to G = {g +
h,h ′ : (h,h′) ∈ H × H} = {g −
h,h ′ :
(h,h ′ ) ∈ H × H}, and noting that S(G,n) ≤ S(H,n) 2 ,
gives the following lemma.
Lemma 2. Let αn = √ (4/n)ln(8S(H,2n) 2 /δ). With
probability at least 1 − δ over an i.i.d. sample Z of size n
from D, we have for all (h,h′) ∈ H × H,
err(h,Z) − err(h ′ ,Z) ≤
errD(h) − errD(h ′ ) + α 2 √
n + αn( EZ[g +
h,h ′]
√
+ EZ[g −
h,h ′]).
With Z = Sn ∪Tn, the error difference on the left-hand side
is errn(h) − errn(h ′ ), which can be empirically determined
because it is equal to êrrn(h) − êrrn(h ′ ). But the terms in
the square root on the right-hand side still pose a problem,
which we fix next.
Corollary 1. Let βn =
√
(4/n)ln(8(n
2
+ n)S(H,2n)
2
/δ). Then, with probability
at least 1 − δ, for all n ≥ 1 and all (h,h ′ ) ∈ H × H
consistent with ˆ Sn, we have
êrrn(h) − êrrn(h ′ ) ≤
errD(h) − errD(h ′ ) + β 2 n + βn( √ êrrn(h) + √ êrrn(h ′ )).
Proof. For each n ≥ 1, we apply Lemma 2 using Z =
Sn ∪Tn and δ = δ/(n 2 +n). Then, we apply a union bound
over all n ≥ 1. Thus, with probability at least 1 − δ, the
bounds in Lemma 2 hold simultaneously for all n ≥ 1 and
all (h,h ′ ) ∈ H 2 with Sn ∪ Tn in place of Z. The corollary
follows because errn(h) − errn(h ′ ) = êrrn(h) − êrrn(h ′ );
and because ESn∪Tn [g+
h,h ′] ≤ êrrn(h) and ESn∪Tn [g−
h,h ′] ≤
êrrn(h ′). To see the first of these expectation bounds, witness
that because h and h ′ agree on Sn,
ESn∪Tn [g+
h,h ′]
= 1
n
≤ 1
n
∑
(x,y)∈Tn
∑
(x,y)∈Tn
The second bound is similar.
1l[h(x) ̸= y ∧ h ′ (x) = y]
1l[h(x) ̸= y] = êrrn(h).
Corollary 1 implies that we can effectively apply the normalized
uniform convergence bounds from Lemma 1 to empirical
error differences on ˆ Sn ∪ Tn, even though ˆ Sn ∪ Tn
is not an i.i.d. sample from D. In light of this, we use the
following setting of ∆n:
(√
êrrn(h+1) + √ )
êrrn(h−1) (1)
∆n := β 2 n + βn
where βn = √ (4/n)ln(8(n 2 + n)S(H,2n) 2 /δ) =
Õ( √ d log n/n) as per Corollary 1.
Correctness and fall-back analysis
We now justify our setting of ∆n with a correctness proof
and fall-back guarantee.
The following lemma elucidates how the inferred labels in
ˆS serve as a mechanism for implicitly maintaining a candidate
set of hypotheses that always includes h ∗ . The fall-back
guarantee then follows almost immediately.
Lemma 3. With probability at least 1 − δ, the hypothesis
h ∗ = arg infh∈H errD(h) is consistent with ˆ Sn for all n ≥ 0
in Algorithm 1.
Proof. Apply the bounds in Corollary 1 (they hold with
probability at least 1 − δ) and proceed by induction on n.
The base case is trivial since ˆ S0 = ∅. Now assume h∗ is consistent with ˆ Sn. Suppose upon receiving xn+1, we
discover êrrn(h+1) − êrrn(h−1) > ∆n. We will show
that h∗ (xn+1) = −1 (assume both h+1 and h−1 exist,
since it is clear h∗ (xn+1) = −1 if h+1 does not exist).
Suppose for the sake of contradiction that h∗ (xn+1) =
+1. We know that êrrn(h∗ ) ≥ êrrn(h+1) (by the inductive
hypothesis h∗ is consistent with ˆ Sn, and yet the
learner chose h+1 in preference to it) and êrrn(h+1) −
êrrn(h−1) > β2 n + βn( √ êrrn(h+1) + √ êrrn(h−1)). In
particular, êrrn(h+1) > β2 n. Therefore,
êrrn(h ∗ ) − êrrn(h−1)
= (êrrn(h ∗ ) − êrrn(h+1)) + (êrrn(h+1) − êrrn(h−1))
> √ êrrn(h+1)( √ êrrn(h ∗ ) − √ êrrn(h+1))
+ β 2 n + βn( √ êrrn(h+1) + √ êrrn(h−1))
> βn( √ êrrn(h ∗ ) − √ êrrn(h+1))
+ β 2 n + βn( √ êrrn(h+1) + √ êrrn(h−1))
= β 2 n + βn( √ êrrn(h ∗ ) + √ êrrn(h−1)).
Now Corollary 1 implies that errD(h ∗ ) > errD(h−1), a contradiction.
Theorem 1. Let ν = infh∈H errD(h) and d = vcdim(H).
There exists a constant c > 0 such that the following holds.
If Algorithm 1 is given a stream of m unlabeled examples,
then with probability at least 1 − δ, the algorithm returns
a hypothesis with error at most ν + c · ((1/m)(d log m +
log(1/δ)) + √ (ν/m)(d log m + log(1/δ))).
Proof. Lemma 3 implies that h∗ is consistent with ˆ Sm with
probability at least 1 − δ. Using the same bounds from
Corollary 1 (already applied in Lemma 3) on h∗ and hf
together with the fact êrrm(hf) ≤ êrrm(h∗ ), we have
errD(hf) ≤ ν + β2 √ √
m + βm ν + βm errD(hf), which in
√
ν.
turn implies errD(hf) ≤ ν + 3β 2 m + 2βm
So, Algorithm 1 returns a hypothesis with error at most ν+ε
when m = Õ((d/ε)(1 + ν/ε)); this is (asymptotically) the
usual sample complexity of supervised learning. Since thealgorithm requests at most m labels, its label complexity is
always at most Õ((d/ε)(1 + ν/ε)).
Label complexity analysis
We can also bound the label complexity of our algorithm in
terms of the disagreement coefficient θ. This yields tighter
bounds when θ is bounded independently of 1/(ε + ν). The
key to deriving our label complexity bounds based on θ is
noting that the probability of requesting the (n + 1)st label
is intimately related to θ and ∆n.
Lemma 4. There exist constants c1,c2 > 0 such that, with
probability at least 1−2δ, for all n ≥ 1, the following holds.
Let h ∗ (xn+1) = ˆy where h ∗ = arg infh∈H errD(h). Then,
the probability that Algorithm 1 requests the label yn+1 is
Prxn+1∼DX
[Request yn+1]
≤ Prxn+1∼DX [errD(h−ˆy) ≤ c1ν + c2β 2 n]
where βn is as defined in Corollary 1 and ν =
infh∈H errD(h).
Proof. See full version of paper.
Lemma 5. In the same setting as Lemma 4, there exists a
constant c > 0 such that Prxn+1∼DX [Request yn+1] ≤ c ·
θ · (ν + β2 n), where θ = θ(D, H,3β 2 √
m + 2βm ν) is the
disagreement coefficient, ν = infh∈H errD(h), and βn is as
defined in Corollary 1.
Proof. Suppose h ∗ (xn+1) = −1. By the triangle inequality,
we have that errD(h+1) ≥ ρ(h+1,h ∗ ) − ν, where ρ is the
disagreement metric on H (Definition 1). By Lemma 4, this
implies that the probability of requesting yn+1 is at most
the probability that ρ(h+1,h ∗ ) ≤ (c1 + 1)ν + c2β 2 n for
some constants c1,c2 > 0. We can choose the constants
so that (c1 + 1)ν + c2β2 n ≥ ν + 3β2 √
m + 2βm ν. Then, the
definition of the disagreement coefficient gives the conclusion
that Prxn+1∼DX [ρ(h+1,h ∗ ) ≤ (c1 + 1)ν + c2β2 n] ≤
θ · ((c1 + 1)ν + c2β2 n).
Now we give our main label complexity bound for agnostic
active learning.
Theorem 2. Let m be the number of unlabeled data given
to Algorithm 1, d = vcdim(H), ν = infh∈H errD(h), βm
as defined in Corollary 1, and θ = θ(D, H,3β 2 √
m+2βm ν).
There exists a constant c1 > 0 such that for any c2 ≥ 1, with
probability at least 1 − 2δ:
1. If ν ≤ (c2 − 1)β2 m, Algorithm 1 returns a hypothesis with
error as bounded in Theorem 1 and the expected number
of labels requested is at most
(
1 + c1c2θ · d log 2 m + log 1
)
log m .
δ
2. Else, the same holds except the expected number of labels
requested is at most
(
1 + c1θ · νm + d log 2 m + log 1
)
log m .
δ
Furthermore, if L is the expected number of labels requested
as per above, then with probability at least 1 − δ ′ , the algorithm
requests no more than L + √ 3Llog(1/δ ′ ) labels.
Proof. Follows from Lemma 5 and a Chernoff bound for the
Poisson trials 1l[Request yn].
With the substitution ε = 3β2 √
m + 2βm ν as per Theorem
1, Theorem 2 entails that for any hypothesis class
and data distribution for which the disagreement coefficient
θ = θ(D, H,ε) is bounded independently of 1/(ε + ν)
(see (Hanneke, 2007) for some examples), Algorithm 1 only
needs Õ(θd log2 (1/ε)) labels to achieve error ε ≈ ν and
Õ(θd(log 2 (1/ε) + (ν/ε) 2 )) labels to achieve error ε ≪ ν.
The latter matches the dependence on ν/ε in the Ω((ν/ε) 2 )
lower bound (Kääriäinen, 2006).
The linear dependence on θ improves on the quadratic dependence
shown for A 2 (Hanneke, 2007) 4 . For an illustrative
consequence of this, suppose DX is the uniform distribution
on the sphere in R d and H is homogeneous linear separators;
in this case, θ = Θ( √ d). Then the label complexity
of A 2 depends at least quadratically on the dimension,
whereas the corresponding dependence for our algorithm is
d 3/2 . A specially-designed setting of ∆n (say, specific to
the input distribution and hypothesis class) may be able to
further reduce the dependence to d (see Balcan, Broder, and
Zhang (2007)).
Experiments
We implemented Algorithm 1 in a few simple cases to experimentally
demonstrate the label complexity improvements.
In each case, the data distribution DX was uniform over
[0,1]; the stream length was m = 10000, and each experiment
was repeated 20 times with different random seeds.
Our first experiment studied linear thresholds on the line.
The target hypothesis was fixed to be h ∗ (x) = sign(x−0.5).
For this hypothesis class, we used two different noise models,
each of which ensured infh∈H errD(h) = errD(h ∗ ) = ν
for a pre-specified ν ∈ [0,1]. The first model was random
misclassification: for each point x ∼ DX , we independently
labeled it h ∗ (x) with probability 1 − ν and −h ∗ (x) with
probability ν. In the second model (also used in Castro and
Nowak (2006)), for each point x ∼ DX , we independently
labeled it +1 with probability (x − 0.5)/(4ν) + 0.5 and −1
otherwise, thus concentrating the noise near the boundary.
Our second experiment studied intervals on the line. Here,
we only used random misclassification, but we varied the
target interval length p+ = Prx∼DX [h∗ (x) = +1].
4
It may be possible to reduce A
2
’s quadratic dependence to a
linear dependence by using normalized bounds, as we do here.The results show that the number of labels requested by Algorithm
1 was exponentially smaller than the total number of
data seen (m) under the first noise model, and was polynomially
smaller under the second noise model (see Figure 2;
we verified the polynomial vs. exponential distinction on
separate log-log scale plots). In the case of intervals, we
observe an initial phase (of duration roughly ∝ 1/p+) in
which every label is requested, followed by a more efficient
phase, confirming the known active-learnability of this class.
These improvements show that our algorithm needed significantly
fewer labels to achieve the same error as a standard
supervised algorithm that uses labels for all points seen.
As a sanity check, we examined the locations of data for
which Algorithm 1 requested a label. We looked at two
particular runs of the algorithm: the first was with H =
intervals, p+ = 0.2, m = 10000, and ν = 0.1; the second
was with H = boxes (d = 2), p+ = 0.49, m = 1000, and
ν = 0.01. In each case, the data distribution was uniform
over [0,1] d , and the noise model was random misclassification.
Figure 3 shows that, early on, labels were requested
everywhere. But as the algorithm progressed, label requests
concentrated near the boundary of the target hypothesis.
Conclusion and future work
We have presented a simple and natural approach to agnostic
active learning. Our extension of the selective sampling
scheme of Cohn-Atlas-Ladner (1994)
1. simplifies the maintenance of the region of uncertainty
with a reduction to supervised learning, and
2. guards against noise with a suitable algorithmic application
of generalization bounds.
Our algorithm relies on a threshold parameter ∆n for comparing
empirical errors. We prescribe a very simple and
natural choice for ∆n – a normalized generalization bound
from supervised learning – but one could hope for a more
clever or aggressive choice, akin to those in Balcan, Broder,
and Zhang (2007) for linear separators.
Finding consistent hypotheses when data is separable is
often a simple task. In such cases, reduction-based active
learning algorithms can be relatively efficient. On the
other hand, agnostic supervised learning is computationally
intractable for many hypothesis classes (Guruswami and
Raghavendra, 2006), and of course, agnostic active learning
is at least as hard in the worst case. Our reduction to supervised
learning is benign in the sense that the learning problems
we need to solve are over samples from the original distribution,
so we do not create pathologically hard instances
(like those arising from hardness reductions) unless they are
inherent in the data. Nevertheless, an important research direction
is to develop consistent active learning algorithms
that only require solving tractable (e.g. convex) optimization
problems. A similar reduction-based scheme may be
possible.
Acknowledgements
We are grateful to the Engineering Institute (a research and
educational partnership between Los Alamos National Laboratory
and U.C. San Diego) for supporting the second author
with a graduate fellowship, and to the NSF for support
under grants IIS-0347646 and IIS-0713540.
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Figure 2: Labeling rate plots. The plots show the number of labels requested (vertical axis) versus the total number of points
seen (labeled + unlabeled, horizontal axis) using Algorithm 1. (a) H = thresholds: under random misclassification noise
with ν = 0 (solid), 0.1 (dashed), 0.2 (dot-dashed); under the boundary noise model with ν = 0.1 (lower dotted), 0.2 (upper
dotted). (b) H = intervals: under random misclassification with (p+,ν) = (0.2,0.0) (solid), (0.1,0.0) (dashed), (0.2,0.1)
(dot-dashed), (0.1,0.1) (dotted).
(a)
(b)
Figure 3: Locations of label requests. (a) H = intervals, h ∗
= [0.4,0.6]. The top histogram shows the locations of first
400 label requests (the x-axis is the unit interval); the bottom histogram is for all (2141) label requests. (b) H = boxes,
h ∗ = [0.15,0.85] 2 . The first 200 requests occurred at the ×s, the next 200 at the ▽s, and the final 109 at the ○s.