48 research outputs found
Online Matrix Completion with Side Information
We give an online algorithm and prove novel mistake and regret bounds for
online binary matrix completion with side information. The mistake bounds we
prove are of the form . The term is
analogous to the usual margin term in SVM (perceptron) bounds. More
specifically, if we assume that there is some factorization of the underlying
matrix into where the rows of are interpreted
as "classifiers" in and the rows of as "instances" in
, then is the maximum (normalized) margin over all
factorizations consistent with the observed matrix. The
quasi-dimension term measures the quality of side information. In the
presence of vacuous side information, . However, if the side
information is predictive of the underlying factorization of the matrix, then
in an ideal case, where is the number of distinct row
factors and is the number of distinct column factors. We additionally
provide a generalization of our algorithm to the inductive setting. In this
setting, we provide an example where the side information is not directly
specified in advance. For this example, the quasi-dimension is now bounded
by
Improved Regret Bounds for Tracking Experts with Memory
We address the problem of sequential prediction with expert advice in a
non-stationary environment with long-term memory guarantees in the sense of
Bousquet and Warmuth [4]. We give a linear-time algorithm that improves on the
best known regret bounds [26]. This algorithm incorporates a relative entropy
projection step. This projection is advantageous over previous weight-sharing
approaches in that weight updates may come with implicit costs as in for
example portfolio optimization. We give an algorithm to compute this projection
step in linear time, which may be of independent interest
On similarity prediction and pairwise clustering
We consider the problem of clustering a finite set of items from pairwise similarity information. Unlike what is done in the literature on this subject, we do so in a passive learning setting, and with no specific constraints on the cluster shapes other than their size. We investigate the problem in different settings: i. an online setting, where we provide a tight characterization of the prediction complexity in the mistake bound model, and ii. a standard stochastic batch setting, where we give tight upper and lower bounds on the achievable generalization error. Prediction performance is measured both in terms of the ability to recover the similarity function encoding the hidden clustering and in terms of how well we classify each item within the set. The proposed algorithms are time efficient
MaxHedge: Maximising a Maximum Online
We introduce a new online learning framework where, at each trial, the
learner is required to select a subset of actions from a given known action
set. Each action is associated with an energy value, a reward and a cost. The
sum of the energies of the actions selected cannot exceed a given energy
budget. The goal is to maximise the cumulative profit, where the profit
obtained on a single trial is defined as the difference between the maximum
reward among the selected actions and the sum of their costs. Action energy
values and the budget are known and fixed. All rewards and costs associated
with each action change over time and are revealed at each trial only after the
learner's selection of actions. Our framework encompasses several online
learning problems where the environment changes over time; and the solution
trades-off between minimising the costs and maximising the maximum reward of
the selected subset of actions, while being constrained to an action energy
budget. The algorithm that we propose is efficient and general in that it may
be specialised to multiple natural online combinatorial problems.Comment: Published in AISTATS 201
Online Multitask Learning with Long-Term Memory
We introduce a novel online multitask setting. In this setting each task is
partitioned into a sequence of segments that is unknown to the learner.
Associated with each segment is a hypothesis from some hypothesis class. We
give algorithms that are designed to exploit the scenario where there are many
such segments but significantly fewer associated hypotheses. We prove regret
bounds that hold for any segmentation of the tasks and any association of
hypotheses to the segments. In the single-task setting this is equivalent to
switching with long-term memory in the sense of [Bousquet and Warmuth; 2003].
We provide an algorithm that predicts on each trial in time linear in the
number of hypotheses when the hypothesis class is finite. We also consider
infinite hypothesis classes from reproducing kernel Hilbert spaces for which we
give an algorithm whose per trial time complexity is cubic in the number of
cumulative trials. In the single-task special case this is the first example of
an efficient regret-bounded switching algorithm with long-term memory for a
non-parametric hypothesis class