8 research outputs found
A Weighted Version of Erdős-Kac Theorem
Let denote the number of distinct prime factors of a natural number . A celebrated result of Erd{\H o}s and Kac states that as a Gaussian distribution. In this thesis, we establish a weighted version of Erd{\H o}s-Kac Theorem. Specifically, we show that the Gaussian limiting distribution is preserved, but shifted, when is weighted by the fold divisor function . We establish this result by computing all positive integral moments of weighted by .
We also provide a proof of the classical identity of for using Dirichlet\u27s kernel
A Characterization of Online Multiclass Learnability
We consider the problem of online multiclass learning when the number of
labels is unbounded. We show that the Multiclass Littlestone dimension, first
introduced in \cite{DanielyERMprinciple}, continues to characterize online
learnability in this setting. Our result complements the recent work by
\cite{Brukhimetal2022} who give a characterization of batch multiclass
learnability when the label space is unbounded.Comment: 8 page
Online Infinite-Dimensional Regression: Learning Linear Operators
We consider the problem of learning linear operators under squared loss
between two infinite-dimensional Hilbert spaces in the online setting. We show
that the class of linear operators with uniformly bounded -Schatten norm is
online learnable for any . On the other hand, we prove an
impossibility result by showing that the class of uniformly bounded linear
operators with respect to the operator norm is \textit{not} online learnable.
Moreover, we show a separation between online uniform convergence and online
learnability by identifying a class of bounded linear operators that is online
learnable but uniform convergence does not hold. Finally, we prove that the
impossibility result and the separation between uniform convergence and
learnability also hold in the agnostic PAC setting.Comment: 17 page
A Characterization of Multioutput Learnability
We consider the problem of learning multioutput function classes in batch and
online settings. In both settings, we show that a multioutput function class is
learnable if and only if each single-output restriction of the function class
is learnable. This provides a complete characterization of the learnability of
multilabel classification and multioutput regression in both batch and online
settings. As an extension, we also consider multilabel learnability in the
bandit feedback setting and show a similar characterization as in the
full-feedback setting.Comment: 37, Updated Online Sectio
Revisiting the Learnability of Apple Tasting
In online binary classification under \textit{apple tasting} feedback, the
learner only observes the true label if it predicts "1". First studied by
\cite{helmbold2000apple}, we revisit this classical partial-feedback setting
and study online learnability from a combinatorial perspective. We show that
the Littlestone dimension continues to prove a tight quantitative
characterization of apple tasting in the agnostic setting, closing an open
question posed by \cite{helmbold2000apple}. In addition, we give a new
combinatorial parameter, called the Effective width, that tightly quantifies
the minimax expected mistakes in the realizable setting. As a corollary, we use
the Effective width to establish a \textit{trichotomy} of the minimax expected
number of mistakes in the realizable setting. In particular, we show that in
the realizable setting, the expected number of mistakes for any learner under
apple tasting feedback can only be , or
.Comment: 18 page
Multiclass Online Learnability under Bandit Feedback
We study online multiclass classification under bandit feedback. We extend
the results of Daniely and Helbertal [2013] by showing that the finiteness of
the Bandit Littlestone dimension is necessary and sufficient for bandit online
multiclass learnability even when the label space is unbounded. Moreover, we
show that, unlike the full-information setting, sequential uniform convergence
is necessary but not sufficient for bandit online learnability. Our result
complements the recent work by Hanneke, Moran, Raman, Subedi, and Tewari [2023]
who show that the Littlestone dimension characterizes online multiclass
learnability in the full-information setting even when the label space is
unbounded.Comment: 11 page
Online Learning with Set-Valued Feedback
We study a variant of online multiclass classification where the learner
predicts a single label but receives a \textit{set of labels} as feedback. In
this model, the learner is penalized for not outputting a label contained in
the revealed set. We show that unlike online multiclass learning with
single-label feedback, deterministic and randomized online learnability are
\textit{not equivalent} in the realizable setting under set-valued feedback. In
addition, we show that deterministic and randomized realizable learnability are
equivalent if the Helly number of the collection of sets that can be revealed
as feedback is finite. In light of this separation, we give two new
combinatorial dimensions, named the Set Littlestone and Measure Shattering
dimension, whose finiteness characterizes deterministic and randomized
realizable learnability respectively. Additionally, these dimensions lower- and
upper bound the deterministic and randomized minimax regret in the realizable
setting. Going beyond the realizable setting, we prove that the Measure
shattering dimension continues to characterize learnability and quantify
minimax regret in the agnostic setting. Finally, we use our results to
establish bounds on the minimax regret for three practical learning settings:
online multilabel ranking, online multilabel classification, and real-valued
prediction with interval-valued response.Comment: 31 page
Sums of random multiplicative functions over function fields with few irreducible factors
We establish a normal approximation for the limiting distribution of partial
sums of random Rademacher multiplicative functions over function fields,
provided the number of irreducible factors of the polynomials is small enough.
This parallels work of Harper for random Rademacher multiplicative functions
over the integers.Comment: 10 pages. Simplification of the proof of Lemma 5 and typos corrected,
one reference adde