arXiv subjects

Hakan Gokcesu

Publications and source records attributed to Hakan Gokcesu.

At least 37 records · Page 2Linked to original sources

arXiv2022

A Linearithmic Time Locally Optimal Algorithm for the Multiway Number Partition Optimization

We study the problem of multiway number partition optimization, which has a myriad of applications in the decision, learning and optimization literature. Even though the original multiway partitioning problem is NP-hard and requires exponential time complexity algorithms; we formulate an easier optimization problem, where our goal is to find a solution that is locally optimal. We propose a linearithmic time complexity $O(N\log N)$ algorithm that can produce such a locally optimal solution. Our method is robust against the input and requires neither positive nor integer inputs.

cs.DS↗

arXiv2022

Smoothing with the Best Rectangle Window is Optimal for All Tapered Rectangle Windows

We investigate the optimal selection of weight windows for the problem of weighted least squares. We show that weight windows should be symmetric around its center, which is also its peak. We consider the class of tapered rectangle window weights, which are nonincreasing away from the center. We show that the best rectangle window is optimal for such window definitions. We also extend our results to the least absolutes and more general case of arbitrary loss functions to find similar results.

stat.ML↗

arXiv2022

Nonconvex Extension of Generalized Huber Loss for Robust Learning and Pseudo-Mode Statistics

We propose an extended generalization of the pseudo Huber loss formulation. We show that using the log-exp transform together with the logistic function, we can create a loss which combines the desirable properties of the strictly convex losses with robust loss functions. With this formulation, we show that a linear convergence algorithm can be utilized to find a minimizer. We further discuss the creation of a quasi-convex composite loss and provide a derivative-free exponential convergence rate algorithm.

stat.ML↗

arXiv2022

Low Regret Binary Sampling Method for Efficient Global Optimization of Univariate Functions

In this work, we propose a computationally efficient algorithm for the problem of global optimization in univariate loss functions. For the performance evaluation, we study the cumulative regret of the algorithm instead of the simple regret between our best query and the optimal value of the objective function. Although our approach has similar regret results with the traditional lower-bounding algorithms such as the Piyavskii-Shubert method for the Lipschitz continuous or Lipschitz smooth functions, it has a major computational cost advantage. In Piyavskii-Shubert method, for certain types of functions, the query points may be hard to determine (as they are solutions to additional optimization problems). However, this issue is circumvented in our binary sampling approach, where the sampling set is predetermined irrespective of the function characteristics. For a search space of $[0,1]$, our approach has at most $L\log (3T)$ and $2.25H$ regret for $L$-Lipschitz continuous and $H$-Lipschitz smooth functions respectively. We also analytically extend our results for a broader class of functions that covers more complex regularity conditions.

cs.LG↗

arXiv2022

QoE Evaluation for Adaptive Video Streaming: Enhanced MDT with Deep Learning

The network performance is usually assessed by drive tests, where teams of people with specially equipped vehicles physically drive out to test various locations throughout a radio network. However, intelligent and autonomous troubleshooting is considered a crucial enabler for 5G- and 6G-networks. In this paper, we propose an architecture for performing virtual drive tests by facilitating radio-quality data from the user equipment. Our architecture comprises three main components: i) a pattern recognizer that learns a typical pattern for the application from application Key Performance Indicators (KPI); ii) a predictor for mapping network KPI with the application KPI; iii) an anomaly detector that compares the predicted application performance with that of the typical application pattern. In this work, we use a commercial state-of-the-art network optimization tool to collect network and application KPI at different geographical locations and at various times of the day for training an initial learning model. We perform extensive numerical analysis to demonstrate key parameters impacting correct video quality prediction and anomaly detection. We show that the playback time is the single most important parameter affecting the video quality, since video packets are usually buffered ahead of time during the playback. However, radio frequency (RF) performance indicators characterizing the quality of the cellular connection improve the QoE estimation in exceptional cases. We demonstrate the efficacy of our approach by showing that the mean maximum F1-score of our method is 77%. Finally, the proposed architecture is flexible and autonomous, and it can operate with different user applications as long as the relevant user-based traces are available.

cs.NI↗

arXiv2021

Efficient, Anytime Algorithms for Calibration with Isotonic Regression under Strictly Convex Losses

We investigate the calibration of estimations to increase performance with an optimal monotone transform on the estimator outputs. We start by studying the traditional square error setting with its weighted variant and show that the optimal monotone transform is in the form of a unique staircase function. We further show that this staircase behavior is preserved for general strictly convex loss functions. Their optimal monotone transforms are also unique, i.e., there exist a single staircase transform that achieves the minimum loss. We propose a linear time and space algorithm that can find such optimal transforms for specific loss settings. Our algorithm has an online implementation where the optimal transform for the samples observed so far are found in linear space and amortized time when the samples arrive in an ordered fashion. We also extend our results to cases where the functions are not trivial to individually optimize and propose an anytime algorithm, which has linear space and pseudo-linearithmic time complexity.

cs.LG↗

arXiv2021

Near-Linear Time Algorithm with Near-Logarithmic Regret Per Switch for Mixable/Exp-Concave Losses

We investigate the problem of online learning, which has gained significant attention in recent years due to its applicability in a wide range of fields from machine learning to game theory. Specifically, we study the online optimization of mixable loss functions with logarithmic static regret in a dynamic environment. The best dynamic estimation sequence that we compete against is selected in hindsight with full observation of the loss functions and is allowed to select different optimal estimations in different time intervals (segments). We propose an online mixture framework that uses these static solvers as the base algorithm. We show that with the suitable selection of hyper-expert creations and weighting strategies, we can achieve logarithmic and squared logarithmic regret per switch in quadratic and linearithmic computational complexity, respectively. For the first time in literature, we show that it is also possible to achieve near-logarithmic regret per switch with sub-polynomial complexity per time. Our results are guaranteed to hold in a strong deterministic sense in an individual sequence manner.

cs.LG↗

arXiv2021

Generalized Translation and Scale Invariant Online Algorithm for Adversarial Multi-Armed Bandits

We study the adversarial multi-armed bandit problem and create a completely online algorithmic framework that is invariant under arbitrary translations and scales of the arm losses. We study the expected performance of our algorithm against a generic competition class, which makes it applicable for a wide variety of problem scenarios. Our algorithm works from a universal prediction perspective and the performance measure used is the expected regret against arbitrary arm selection sequences, which is the difference between our losses and a competing loss sequence. The competition class can be designed to include fixed arm selections, switching bandits, contextual bandits, or any other competition of interest. The sequences in the competition class are generally determined by the specific application at hand and should be designed accordingly. Our algorithm neither uses nor needs any preliminary information about the loss sequences and is completely online. Its performance bounds are the second order bounds in terms of sum of the squared losses, where any affine transform of the losses has no effect on the normalized regret.

cs.LG↗

arXiv2021

A Quadratic Time Locally Optimal Algorithm for NP-hard Equal Cardinality Partition Optimization

We study the optimization version of the equal cardinality set partition problem (where the absolute difference between the equal sized partitions' sums are minimized). While this problem is NP-hard and requires exponential complexity to solve in general, we have formulated a weaker version of this NP-hard problem, where the goal is to find a locally optimal solution. The local optimality considered in our work is under any swap between the opposing partitions' element pairs. To this end, we designed an algorithm which can produce such a locally optimal solution in $O(N^2)$ time and $O(N)$ space. Our approach does not require positive or integer inputs and works equally well under arbitrary input precisions. Thus, it is widely applicable in different problem scenarios.

cs.DS↗

arXiv2021

Efficient Locally Optimal Number Set Partitioning for Scheduling, Allocation and Fair Selection

We study the optimization version of the set partition problem (where the difference between the partition sums are minimized), which has numerous applications in decision theory literature. While the set partitioning problem is NP-hard and requires exponential complexity to solve (i.e., intractable); we formulate a weaker version of this NP-hard problem, where the goal is to find a locally optimal solution. We show that our proposed algorithms can find a locally optimal solution in near linear time. Our algorithms require neither positive nor integer elements in the input set, hence, they are more widely applicable.

cs.DS↗

arXiv2021

Nonparametric Extrema Analysis in Time Series for Envelope Extraction, Peak Detection and Clustering

In this paper, we propose a nonparametric approach that can be used in envelope extraction, peak-burst detection and clustering in time series. Our problem formalization results in a naturally defined splitting/forking of the time series. With a possibly hierarchical implementation, it can be used for various applications in machine learning, signal processing and mathematical finance. From an incoming input signal, our iterative procedure sequentially creates two signals (one upper bounding and one lower bounding signal) by minimizing the cumulative $L_1$ drift. We show that a solution can be efficiently calculated by use of a Viterbi-like path tracking algorithm together with an optimal elimination rule. We consider many interesting settings, where our algorithm has near-linear time complexities.

cs.LG↗

arXiv2021

Generalized Huber Loss for Robust Learning and its Efficient Minimization for a Robust Statistics

We propose a generalized formulation of the Huber loss. We show that with a suitable function of choice, specifically the log-exp transform; we can achieve a loss function which combines the desirable properties of both the absolute and the quadratic loss. We provide an algorithm to find the minimizer of such loss functions and show that finding a centralizing metric is not that much harder than the traditional mean and median.

stat.ML↗

arXiv2021

Optimally Efficient Sequential Calibration of Binary Classifiers to Minimize Classification Error

In this work, we aim to calibrate the score outputs of an estimator for the binary classification problem by finding an 'optimal' mapping to class probabilities, where the 'optimal' mapping is in the sense that minimizes the classification error (or equivalently, maximizes the accuracy). We show that for the given target variables and the score outputs of an estimator, an 'optimal' soft mapping, which monotonically maps the score values to probabilities, is a hard mapping that maps the score values to $0$ and $1$. We show that for class weighted (where the accuracy for one class is more important) and sample weighted (where the samples' accurate classifications are not equally important) errors, or even general linear losses; this hard mapping characteristic is preserved. We propose a sequential recursive merger approach, which produces an 'optimal' hard mapping (for the observed samples so far) sequentially with each incoming new sample. Our approach has a logarithmic in sample size time complexity, which is optimally efficient.

cs.LG↗

arXiv2021

Optimal and Efficient Algorithms for General Mixable Losses against Switching Oracles

We investigate the problem of online learning, which has gained significant attention in recent years due to its applicability in a wide range of fields from machine learning to game theory. Specifically, we study the online optimization of mixable loss functions in a dynamic environment. We introduce online mixture schemes that asymptotically achieves the performance of the best dynamic estimation sequence of the switching oracle with optimal regret redundancies. The best dynamic estimation sequence that we compete against is selected in hindsight with full observation of the loss functions and is allowed to select different optimal estimations in different time intervals (segments). We propose two mixtures in our work. Firstly, we propose a tractable polynomial time complexity algorithm that can achieve the optimal redundancy of the intractable brute force approach. Secondly, we propose an efficient logarithmic time complexity algorithm that can achieve the optimal redundancy up to a constant multiplicity gap. Our results are guaranteed to hold in a strong deterministic sense in an individual sequence manner.

cs.LG↗

arXiv2020

Recursive Experts: An Efficient Optimal Mixture of Learning Systems in Dynamic Environments

Sequential learning systems are used in a wide variety of problems from decision making to optimization, where they provide a 'belief' (opinion) to nature, and then update this belief based on the feedback (result) to minimize (or maximize) some cost or loss (conversely, utility or gain). The goal is to reach an objective by exploiting the temporal relation inherent to the nature's feedback (state). By exploiting this relation, specific learning systems can be designed that perform asymptotically optimal for various applications. However, if the framework of the problem is not stationary, i.e., the nature's state sometimes changes arbitrarily, the past cumulative belief revision done by the system may become useless and the system may fail if it lacks adaptivity. While this adaptivity can be directly implemented in specific cases (e.g., convex optimization), it is mostly not straightforward for general learning tasks. To this end, we propose an efficient optimal mixture framework for general sequential learning systems, which we call the recursive experts for dynamic environments. For this purpose, we design hyper-experts that incorporate the learning systems at our disposal and recursively merge in a specific way to achieve minimax optimal regret bounds up to constant factors. The multiplicative increases in computational complexity from the initial system to our adaptive system are only logarithmic-in-time factors.

cs.LG↗

arXiv2020

A Generalized Online Algorithm for Translation and Scale Invariant Prediction with Expert Advice

In this work, we aim to create a completely online algorithmic framework for prediction with expert advice that is translation-free and scale-free of the expert losses. Our goal is to create a generalized algorithm that is suitable for use in a wide variety of applications. For this purpose, we study the expected regret of our algorithm against a generic competition class in the sequential prediction by expert advice problem, where the expected regret measures the difference between the losses of our prediction algorithm and the losses of the 'best' expert selection strategy in the competition. We design our algorithm using the universal prediction perspective to compete against a specified class of expert selection strategies, which is not necessarily a fixed expert selection. The class of expert selection strategies that we want to compete against is purely determined by the specific application at hand and is left generic, which makes our generalized algorithm suitable for use in many different problems. We show that no preliminary knowledge about the loss sequence is required by our algorithm and its performance bounds, which are second order, expressed in terms of sums of squared losses. Our regret bounds are stable under arbitrary scalings and translations of the losses.

cs.LG↗

arXiv2019

Minimax Optimal Algorithms for Adversarial Bandit Problem with Multiple Plays

We investigate the adversarial bandit problem with multiple plays under semi-bandit feedback. We introduce a highly efficient algorithm that asymptotically achieves the performance of the best switching $m$-arm strategy with minimax optimal regret bounds. To construct our algorithm, we introduce a new expert advice algorithm for the multiple-play setting. By using our expert advice algorithm, we additionally improve the best-known high-probability bound for the multi-play setting by $O(\sqrt{m})$. Our results are guaranteed to hold in an individual sequence manner since we have no statistical assumption on the bandit arm gains. Through an extensive set of experiments involving synthetic and real data, we demonstrate significant performance gains achieved by the proposed algorithm with respect to the state-of-the-art algorithms.

cs.LG↗

arXiv2019

Accelerating Min-Max Optimization with Application to Minimal Bounding Sphere

We study the min-max optimization problem where each function contributing to the max operation is strongly-convex and smooth with bounded gradient in the search domain. By smoothing the max operator, we show the ability to achieve an arbitrarily small positive optimality gap of $δ$ in $\tilde{O}(1/\sqrtδ)$ computational complexity (up to logarithmic factors) as opposed to the state-of-the-art strong-convexity computational requirement of $O(1/δ)$. We apply this important result to the well-known minimal bounding sphere problem and demonstrate that we can achieve a $(1+\varepsilon)$-approximation of the minimal bounding sphere, i.e. identify an hypersphere enclosing a total of $n$ given points in the $d$ dimensional unbounded space $\mathbb{R}^d$ with a radius at most $(1+\varepsilon)$ times the actual minimal bounding sphere radius for an arbitrarily small positive $\varepsilon$, in $\tilde{O}(n d /\sqrt{\varepsilon})$ computational time as opposed to the state-of-the-art approach of core-set methodology, which needs $O(n d /\varepsilon)$ computational time.

math.OC↗