An efficient algorithm for learning with semi-bandit feedback

We consider the problem of online combinatorial optimization under semi-bandit feedback. The goal of the learner is to sequentially select its actions from a combinatorial decision set so as to minimize its cumulative loss. We propose a learning algorithm for this problem based on combining the Follow-the-Perturbed-Leader (FPL) prediction method with a novel loss estimation procedure called Geometric Resampling (GR). Contrary to previous solutions, the resulting algorithm can be efficiently implemented for any decision set where efficient offline combinatorial optimization is possible at all. Assuming that the elements of the decision set can be described with d-dimensional binary vectors with at most m non-zero entries, we show that the expected regret of our algorithm after T rounds is O(m sqrt(dT log d)). As a side result, we also improve the best known regret bounds for FPL in the full information setting to O(m^(3/2) sqrt(T log d)), gaining a factor of sqrt(d/m) over previous bounds for this algorithm.Comment: submitted to ALT 201

An Efficient Interior-Point Method for Online Convex Optimization

A new algorithm for regret minimization in online convex optimization is described. The regret of the algorithm after $T$ time periods is $O(\sqrt{T \log T})$ - which is the minimum possible up to a logarithmic term. In addition, the new algorithm is adaptive, in the sense that the regret bounds hold not only for the time periods $1,\ldots,T$ but also for every sub-interval $s,s+1,\ldots,t$. The running time of the algorithm matches that of newly introduced interior point algorithms for regret minimization: in $n$-dimensional space, during each iteration the new algorithm essentially solves a system of linear equations of order $n$, rather than solving some constrained convex optimization problem in $n$ dimensions and possibly many constraints

Approximations for Throughput Maximization

In this paper we study the classical problem of throughput maximization. In this problem we have a collection $J$ of $n$ jobs, each having a release time $r_j$, deadline $d_j$, and processing time $p_j$. They have to be scheduled non-preemptively on $m$ identical parallel machines. The goal is to find a schedule which maximizes the number of jobs scheduled entirely in their $[r_j,d_j]$ window. This problem has been studied extensively (even for the case of $m=1$). Several special cases of the problem remain open. Bar-Noy et al. [STOC1999] presented an algorithm with ratio $1-1/(1+1/m)^m$ for $m$ machines, which approaches $1-1/e$ as $m$ increases. For $m=1$, Chuzhoy-Ostrovsky-Rabani [FOCS2001] presented an algorithm with approximation with ratio $1-\frac{1}{e}-\varepsilon$ (for any $\varepsilon>0$). Recently Im-Li-Moseley [IPCO2017] presented an algorithm with ratio $1-1/e-\varepsilon_0$ for some absolute constant $\varepsilon_0>0$ for any fixed $m$. They also presented an algorithm with ratio $1-O(\sqrt{\log m/m})-\varepsilon$ for general $m$ which approaches 1 as $m$ grows. The approximability of the problem for $m=O(1)$ remains a major open question. Even for the case of $m=1$ and $c=O(1)$ distinct processing times the problem is open (Sgall [ESA2012]). In this paper we study the case of $m=O(1)$ and show that if there are $c$ distinct processing times, i.e. $p_j$'s come from a set of size $c$, then there is a $(1-\varepsilon)$-approximation that runs in time $O(n^{mc^7\varepsilon^{-6}}\log T)$, where $T$ is the largest deadline. Therefore, for constant $m$ and constant $c$ this yields a PTAS. Our algorithm is based on proving structural properties for a near optimum solution that allows one to use a dynamic programming with pruning

Dictionary Learning and Tensor Decomposition via the Sum-of-Squares Method

We give a new approach to the dictionary learning (also known as "sparse coding") problem of recovering an unknown $n\times m$ matrix $A$ (for $m \geq n$) from examples of the form $$y = Ax + e,$$ where $x$ is a random vector in $\mathbb R^m$ with at most $\tau m$ nonzero coordinates, and $e$ is a random noise vector in $\mathbb R^n$ with bounded magnitude. For the case $m=O(n)$, our algorithm recovers every column of $A$ within arbitrarily good constant accuracy in time $m^{O(\log m/\log(\tau^{-1}))}$, in particular achieving polynomial time if $\tau = m^{-\delta}$ for any $\delta>0$, and time $m^{O(\log m)}$ if $\tau$ is (a sufficiently small) constant. Prior algorithms with comparable assumptions on the distribution required the vector $x$ to be much sparser---at most $\sqrt{n}$ nonzero coordinates---and there were intrinsic barriers preventing these algorithms from applying for denser $x$. We achieve this by designing an algorithm for noisy tensor decomposition that can recover, under quite general conditions, an approximate rank-one decomposition of a tensor $T$, given access to a tensor $T'$ that is $\tau$-close to $T$ in the spectral norm (when considered as a matrix). To our knowledge, this is the first algorithm for tensor decomposition that works in the constant spectral-norm noise regime, where there is no guarantee that the local optima of $T$ and $T'$ have similar structures. Our algorithm is based on a novel approach to using and analyzing the Sum of Squares semidefinite programming hierarchy (Parrilo 2000, Lasserre 2001), and it can be viewed as an indication of the utility of this very general and powerful tool for unsupervised learning problems