A fundamental problem in numerical analysis and approximation theory is approximating smooth functions by polynomials. A much harder version under recent consideration is to enforce bounds constraints on the approximating polynomial. In this paper, we consider the problem of approximating functions by polynomials whose Bernstein coefficients with respect to a given degree satisfy such bounds, which implies such bounds on the approximant. We frame the problem as an inequality-constrained optimization problem and give an algorithm for finding the Bernstein coefficients of the exact solution. Additionally, our method can be modified slightly to include equality constraints such as mass preservation. It also extends naturally to multivariate polynomials over a simplex.
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Recent development of Deep Reinforcement Learning has demonstrated superior performance of neural networks in solving challenging problems with large or even continuous state spaces. One specific approach is to deploy neural networks to approximate value functions by minimising the Mean Squared Bellman Error function. Despite great successes of Deep Reinforcement Learning, development of reliable and efficient numerical algorithms to minimise the Bellman Error is still of great scientific interest and practical demand. Such a challenge is partially due to the underlying optimisation problem being highly non-convex or using incorrect gradient information as done in Semi-Gradient algorithms. In this work, we analyse the Mean Squared Bellman Error from a smooth optimisation perspective combined with a Residual Gradient formulation. Our contribution is two-fold. First, we analyse critical points of the error function and provide technical insights on the optimisation procure and design choices for neural networks. When the existence of global minima is assumed and the objective fulfils certain conditions we can eliminate suboptimal local minima when using over-parametrised neural networks. We can construct an efficient Approximate Newton's algorithm based on our analysis and confirm theoretical properties of this algorithm such as being locally quadratically convergent to a global minimum numerically. Second, we demonstrate feasibility and generalisation capabilities of the proposed algorithm empirically using continuous control problems and provide a numerical verification of our critical point analysis. We outline the short coming of Semi-Gradients. To benefit from an approximate Newton's algorithm complete derivatives of the Mean Squared Bellman error must be considered during training.
We consider the notions of (i) critical points, (ii) second-order points, (iii) local minima, and (iv) strict local minima for multivariate polynomials. For each type of point, and as a function of the degree of the polynomial, we study the complexity of deciding (1) if a given point is of that type, and (2) if a polynomial has a point of that type. Our results characterize the complexity of these two questions for all degrees left open by prior literature. Our main contributions reveal that many of these questions turn out to be tractable for cubic polynomials. In particular, we present an efficiently-checkable necessary and sufficient condition for local minimality of a point for a cubic polynomial. We also show that a local minimum of a cubic polynomial can be efficiently found by solving semidefinite programs of size linear in the number of variables. By contrast, we show that it is strongly NP-hard to decide if a cubic polynomial has a critical point. We also prove that the set of second-order points of any cubic polynomial is a spectrahedron, and conversely that any spectrahedron is the projection of the set of second-order points of a cubic polynomial. In our final section, we briefly present a potential application of finding local minima of cubic polynomials to the design of a third-order Newton method.
The class of algorithms called Hessian Estimation Evolution Strategies (HE-ESs) update the covariance matrix of their sampling distribution by directly estimating the curvature of the objective function. The approach is practically efficient, as attested by respectable performance on the BBOB testbed, even on rather irregular functions. In this paper we formally prove two strong guarantees for the (1+4)-HE-ES, a minimal elitist member of the family: stability of the covariance matrix update, and as a consequence, linear convergence on all convex quadratic problems at a rate that is independent of the problem instance.
Several issues in machine learning and inverse problems require to generate discrete data, as if sampled from a model probability distribution. A common way to do so relies on the construction of a uniform probability distribution over a set of $N$ points which minimizes the Wasserstein distance to the model distribution. This minimization problem, where the unknowns are the positions of the atoms, is non-convex. Yet, in most cases, a suitably adjusted version of Lloyd's algorithm -- in which Voronoi cells are replaced by Power cells -- leads to configurations with small Wasserstein error. This is surprising because, again, of the non-convex nature of the problem, as well as the existence of spurious critical points. We provide explicit upper bounds for the convergence speed of this Lloyd-type algorithm, starting from a cloud of points sufficiently far from each other. This already works after one step of the iteration procedure, and similar bounds can be deduced, for the corresponding gradient descent. These bounds naturally lead to a modified Poliak-Lojasiewicz inequality for the Wasserstein distance cost, with an error term depending on the distances between Dirac masses in the discrete distribution.
Submodular optimization has numerous applications such as crowdsourcing and viral marketing. In this paper, we study the fundamental problem of non-negative submodular function maximization subject to a $k$-system constraint, which generalizes many other important constraints in submodular optimization such as cardinality constraint, matroid constraint, and $k$-extendible system constraint. The existing approaches for this problem achieve the best-known approximation ratio of $k+2\sqrt{k+2}+3$ (for a general submodular function) based on deterministic algorithmic frameworks. We propose several randomized algorithms that improve upon the state-of-the-art algorithms in terms of approximation ratio and time complexity, both under the non-adaptive setting and the adaptive setting. The empirical performance of our algorithms is extensively evaluated in several applications related to data mining and social computing, and the experimental results demonstrate the superiorities of our algorithms in terms of both utility and efficiency.
In this paper, we consider an online optimization problem over $T$ rounds where at each step $t\in[T]$, the algorithm chooses an action $x_t$ from the fixed convex and compact domain set $\mathcal{K}$. A utility function $f_t(\cdot)$ is then revealed and the algorithm receives the payoff $f_t(x_t)$. This problem has been previously studied under the assumption that the utilities are adversarially chosen monotone DR-submodular functions and $\mathcal{O}(\sqrt{T})$ regret bounds have been derived. We first characterize the class of strongly DR-submodular functions and then, we derive regret bounds for the following new online settings: $(1)$ $\{f_t\}_{t=1}^T$ are monotone strongly DR-submodular and chosen adversarially, $(2)$ $\{f_t\}_{t=1}^T$ are monotone submodular (while the average $\frac{1}{T}\sum_{t=1}^T f_t$ is strongly DR-submodular) and chosen by an adversary but they arrive in a uniformly random order, $(3)$ $\{f_t\}_{t=1}^T$ are drawn i.i.d. from some unknown distribution $f_t\sim \mathcal{D}$ where the expected function $f(\cdot)=\mathbb{E}_{f_t\sim\mathcal{D}}[f_t(\cdot)]$ is monotone DR-submodular. For $(1)$, we obtain the first logarithmic regret bounds. In terms of the second framework, we show that it is possible to obtain similar logarithmic bounds with high probability. Finally, for the i.i.d. model, we provide algorithms with $\tilde{\mathcal{O}}(\sqrt{T})$ stochastic regret bound, both in expectation and with high probability. Experimental results demonstrate that our algorithms outperform the previous techniques in the aforementioned three settings.
The cumulative empirical spectral measure (CESM) $\Phi[\mathbf{A}] : \mathbb{R} \to [0,1]$ of a $n\times n$ symmetric matrix $\mathbf{A}$ is defined as the fraction of eigenvalues of $\mathbf{A}$ less than a given threshold, i.e., $\Phi[\mathbf{A}](x) := \sum_{i=1}^{n} \frac{1}{n} {\large\unicode{x1D7D9}}[ \lambda_i[\mathbf{A}]\leq x]$. Spectral sums $\operatorname{tr}(f[\mathbf{A}])$ can be computed as the Riemann--Stieltjes integral of $f$ against $\Phi[\mathbf{A}]$, so the task of estimating CESM arises frequently in a number of applications, including machine learning. We present an error analysis for stochastic Lanczos quadrature (SLQ). We show that SLQ obtains an approximation to the CESM within a Wasserstein distance of $t \: | \lambda_{\text{max}}[\mathbf{A}] - \lambda_{\text{min}}[\mathbf{A}] |$ with probability at least $1-\eta$, by applying the Lanczos algorithm for $\lceil 12 t^{-1} + \frac{1}{2} \rceil$ iterations to $\lceil 4 ( n+2 )^{-1}t^{-2} \ln(2n\eta^{-1}) \rceil$ vectors sampled independently and uniformly from the unit sphere. We additionally provide (matrix-dependent) a posteriori error bounds for the Wasserstein and Kolmogorov--Smirnov distances between the output of this algorithm and the true CESM. The quality of our bounds is demonstrated using numerical experiments.
Bayesian optimization (BO) is a popular method for optimizing expensive-to-evaluate black-box functions. BO budgets are typically given in iterations, which implicitly assumes each evaluation has the same cost. In fact, in many BO applications, evaluation costs vary significantly in different regions of the search space. In hyperparameter optimization, the time spent on neural network training increases with layer size; in clinical trials, the monetary cost of drug compounds vary; and in optimal control, control actions have differing complexities. Cost-constrained BO measures convergence with alternative cost metrics such as time, money, or energy, for which the sample efficiency of standard BO methods is ill-suited. For cost-constrained BO, cost efficiency is far more important than sample efficiency. In this paper, we formulate cost-constrained BO as a constrained Markov decision process (CMDP), and develop an efficient rollout approximation to the optimal CMDP policy that takes both the cost and future iterations into account. We validate our method on a collection of hyperparameter optimization problems as well as a sensor set selection application.
We analyse adversarial bandit convex optimisation with an adversary that is restricted to playing functions of the form $f_t(x) = g_t(\langle x, \theta\rangle)$ for convex $g_t : \mathbb R \to \mathbb R$ and unknown $\theta \in \mathbb R^d$ that is homogeneous over time. We provide a short information-theoretic proof that the minimax regret is at most $O(d \sqrt{n} \log(n \operatorname{diam}(\mathcal K)))$ where $n$ is the number of interactions, $d$ the dimension and $\operatorname{diam}(\mathcal K)$ is the diameter of the constraint set.
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This paper shows effectiveness of X3SAT in proving P = NP. This is due to the fact that it is easy to check unsatisfiability of a particular truth assignment. A truth assignment leads to some reductions of clauses by means of "exactly-1 disjunction". The reductions result in a conjunction of literals, called the scope of the assignment. It is proved that this particular truth assignment makes the formula unsatisfiable iff the scope_s is unsatisfiable for some s, which is trivial to check. Then, each literal such that its scope is unsatisfiable is removed from the formula. Therefore, the formula is satisfiable iff the scope of every literal is satisfied.
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