Low-Sensitivity Functions from Unambiguous Certificates

We provide new query complexity separations against sensitivity for total Boolean functions: a power $3$ separation between deterministic (and even randomized or quantum) query complexity and sensitivity, and a power $2.22$ separation between certificate complexity and sensitivity. We get these separations by using a new connection between sensitivity and a seemingly unrelated measure called one-sided unambiguous certificate complexity ($UC_{min}$). We also show that $UC_{min}$ is lower-bounded by fractional block sensitivity, which means we cannot use these techniques to get a super-quadratic separation between $bs(f)$ and $s(f)$. We also provide a quadratic separation between the tree-sensitivity and decision tree complexity of Boolean functions, disproving a conjecture of Gopalan, Servedio, Tal, and Wigderson (CCC 2016). Along the way, we give a power $1.22$ separation between certificate complexity and one-sided unambiguous certificate complexity, improving the power $1.128$ separation due to G\"o\"os (FOCS 2015). As a consequence, we obtain an improved $\Omega(\log^{1.22} n)$ lower-bound on the co-nondeterministic communication complexity of the Clique vs. Independent Set problem.Comment: 25 pages. This version expands the results and adds Pooya Hatami and Avishay Tal as author

{Improved Bounds on Fourier Entropy and Min-entropy}

Given a Boolean function $f:\{-1,1\}^n\to \{-1,1\}$, the Fourier distribution assigns probability $\widehat{f}(S)^2$ to $S\subseteq [n]$. The Fourier Entropy-Influence (FEI) conjecture of Friedgut and Kalai asks if there exist a universal constant C>0 such that $H(\hat{f}^2)\leq C Inf(f)$, where $H(\hat{f}^2)$ is the Shannon entropy of the Fourier distribution of $f$ and $Inf(f)$ is the total influence of $f$. 1) We consider the weaker Fourier Min-entropy-Influence (FMEI) conjecture. This asks if $H_{\infty}(\hat{f}^2)\leq C Inf(f)$, where $H_{\infty}(\hat{f}^2)$ is the min-entropy of the Fourier distribution. We show $H_{\infty}(\hat{f}^2)\leq 2C_{\min}^\oplus(f)$, where $C_{\min}^\oplus(f)$ is the minimum parity certificate complexity of $f$. We also show that for every $\epsilon\geq 0$, we have $H_{\infty}(\hat{f}^2)\leq 2\log (\|\hat{f}\|_{1,\epsilon}/(1-\epsilon))$, where $\|\hat{f}\|_{1,\epsilon}$ is the approximate spectral norm of $f$. As a corollary, we verify the FMEI conjecture for the class of read-$k$ $DNF$s (for constant $k$). 2) We show that $H(\hat{f}^2)\leq 2 aUC^\oplus(f)$, where $aUC^\oplus(f)$ is the average unambiguous parity certificate complexity of $f$. This improves upon Chakraborty et al. An important consequence of the FEI conjecture is the long-standing Mansour's conjecture. We show that a weaker version of FEI already implies Mansour's conjecture: is $H(\hat{f}^2)\leq C \min\{C^0(f),C^1(f)\}$?, where $C^0(f), C^1(f)$ are the 0- and 1-certificate complexities of $f$, respectively. 3) We study what FEI implies about the structure of polynomials that 1/3-approximate a Boolean function. We pose a conjecture (which is implied by FEI): no "flat" degree-$d$ polynomial of sparsity $2^{\omega(d)}$ can 1/3-approximate a Boolean function. We prove this conjecture unconditionally for a particular class of polynomials

Separations in Query Complexity Based on Pointer Functions

In 1986, Saks and Wigderson conjectured that the largest separation between deterministic and zero-error randomized query complexity for a total boolean function is given by the function $f$ on $n=2^k$ bits defined by a complete binary tree of NAND gates of depth $k$, which achieves $R_0(f) = O(D(f)^{0.7537\ldots})$. We show this is false by giving an example of a total boolean function $f$ on $n$ bits whose deterministic query complexity is $\Omega(n/\log(n))$ while its zero-error randomized query complexity is $\tilde O(\sqrt{n})$. We further show that the quantum query complexity of the same function is $\tilde O(n^{1/4})$, giving the first example of a total function with a super-quadratic gap between its quantum and deterministic query complexities. We also construct a total boolean function $g$ on $n$ variables that has zero-error randomized query complexity $\Omega(n/\log(n))$ and bounded-error randomized query complexity $R(g) = \tilde O(\sqrt{n})$. This is the first super-linear separation between these two complexity measures. The exact quantum query complexity of the same function is $Q_E(g) = \tilde O(\sqrt{n})$. These two functions show that the relations $D(f) = O(R_1(f)^2)$ and $R_0(f) = \tilde O(R(f)^2)$ are optimal, up to poly-logarithmic factors. Further variations of these functions give additional separations between other query complexity measures: a cubic separation between $Q$ and $R_0$, a $3/2$-power separation between $Q_E$ and $R$, and a 4th power separation between approximate degree and bounded-error randomized query complexity. All of these examples are variants of a function recently introduced by \goos, Pitassi, and Watson which they used to separate the unambiguous 1-certificate complexity from deterministic query complexity and to resolve the famous Clique versus Independent Set problem in communication complexity.Comment: 25 pages, 6 figures. Version 3 improves separation between Q_E and R_0 and updates reference

Investment timing and optimal capacity choice for small hydropower projects

This paper presents a method for assessing small hydropower projects that are subject to uncertain electricity prices. We present a real options-based method with continuous scaling, and we find that there is a unique price limit for initiating the project. If the current electricity price is below this limit it is never optimal to invest, but above this limit investment is made according to the function for optimal size. The connection between the real option and the physical properties of a small hydropower plant is dealt with using a spreadsheet model that performs a technical simulation of the production in a plant, based on all the important choices for such a plant. The main results of the spreadsheet are simulated production size and the investment costs, which are in turn used for finding the value of the real option and the price limit. The method is illustrated on three different Norwegian small hydropower projects.OR in Energy; Real Options; Continuous Scaling; Project Evaluation; Hydropower

Aggregating the monetary aggregates : concepts and issues

An abstract for this article is not available.Monetary policy ; Federal Reserve System

Separations between Combinatorial Measures for Transitive Functions

The role of symmetry in Boolean functions $f:\{0,1\}^n \to \{0,1\}$ has been extensively studied in complexity theory. For example, symmetric functions, that is, functions that are invariant under the action of $S_n$, is an important class of functions in the study of Boolean functions. A function $f:\{0,1\}^n \to \{0,1\}$ is called transitive (or weakly-symmetric) if there exists a transitive group $G$ of $S_n$ such that $f$ is invariant under the action of $G$ - that is the function value remains unchanged even after the bits of the input of $f$ are moved around according to some permutation $\sigma \in G$. Understanding various complexity measures of transitive functions has been a rich area of research for the past few decades. In this work, we study transitive functions in light of several combinatorial measures. We look at the maximum separation between various pairs of measures for transitive functions. Such study for general Boolean functions has been going on for past many years. The best-known results for general Boolean functions have been nicely compiled by Aaronson et. al (STOC, 2021). The separation between a pair of combinatorial measures is shown by constructing interesting functions that demonstrate the separation. But many of the celebrated separation results are via the construction of functions (like "pointer functions" from Ambainis et al. (JACM, 2017) and "cheat-sheet functions" Aaronson et al. (STOC, 2016)) that are not transitive. Hence, we don't have such separation between the pairs of measures for transitive functions. In this paper we show how to modify some of these functions to construct transitive functions that demonstrate similar separations between pairs of combinatorial measures

Pigou’s Dividend versus Ramsey’s Dividend in the Double Dividend Literature

This paper deals with the welfare analysis of green tax reforms. The aims of this paper are to highlight misinterpretations of policy assessments in the double dividend literature, to specify which of the efficiency costs and benefits should be ascribed to each dividend, and then, to propose a definition for the first dividend and the second dividend. We found the Pigou’s dividend more appropriate for policy guidance, in contrast to the Ramsey’s dividend usually found in mainstream literature. Therefore, we take up some authors’ recent claims about the need of unambiguous and operative definitions of these dividends both for empirical purposes, and political advice. Finally, the paper analyzes a green tax reform for the US economy to illustrate the advantages of our definitions for policy assessment. The new definitions proposed in this paper i) overcome some shortcoming of the mainstream current definitions in the literature regarding overestimation of the efficiency costs; and, ii) provide information by themselves and not as a partial view of the whole picture.Double Dividend, Green Tax Reforms, Ramsey’s Dividend, Pigou’s Dividend

Reinforcement Learning for Mixed-Integer Problems Based on MPC

Model Predictive Control has been recently proposed as policy approximation for Reinforcement Learning, offering a path towards safe and explainable Reinforcement Learning. This approach has been investigated for Q-learning and actor-critic methods, both in the context of nominal Economic MPC and Robust (N)MPC, showing very promising results. In that context, actor-critic methods seem to be the most reliable approach. Many applications include a mixture of continuous and integer inputs, for which the classical actor-critic methods need to be adapted. In this paper, we present a policy approximation based on mixed-integer MPC schemes, and propose a computationally inexpensive technique to generate exploration in the mixed-integer input space that ensures a satisfaction of the constraints. We then propose a simple compatible advantage function approximation for the proposed policy, that allows one to build the gradient of the mixed-integer MPC-based policy.Comment: Accepted at IFAC 202