Smooth markets: A basic mechanism for organizing gradient-based learners

With the success of modern machine learning, it is becoming increasingly important to understand and control how learning algorithms interact. Unfortunately, negative results from game theory show there is little hope of understanding or controlling general n-player games. We therefore introduce smooth markets (SM-games), a class of n-player games with pairwise zero sum interactions. SM-games codify a common design pattern in machine learning that includes (some) GANs, adversarial training, and other recent algorithms. We show that SM-games are amenable to analysis and optimization using first-order methods.Comment: 18 pages, 3 figure

Cadherin-Dependent Cell Morphology in an Epithelium: Constructing a Quantitative Dynamical Model

Cells in the Drosophila retina have well-defined morphologies that are attained during tissue morphogenesis. We present a computer simulation of the epithelial tissue in which the global interfacial energy between cells is minimized. Experimental data for both normal cells and mutant cells either lacking or misexpressing the adhesion protein N-cadherin can be explained by a simple model incorporating salient features of morphogenesis that include the timing of N-cadherin expression in cells and its temporal relationship to the remodeling of cell-cell contacts. The simulations reproduce the geometries of wild-type and mutant cells, distinguish features of cadherin dynamics, and emphasize the importance of adhesion protein biogenesis and its timing with respect to cell remodeling. The simulations also indicate that N-cadherin protein is recycled from inactive interfaces to active interfaces, thereby modulating adhesion strengths between cells

Error measures for N-cadherin knockout mutants.

<p>Error measures for N-cadherin knockout mutants.</p

Destruction and Recycling models for N-cadherin distribution.

<p>Cone cells with unpaired N-cadherin (green), or paired N-cadherin (red) along active interfaces (dashed orange). (<b>A</b>) Distribution and abundance of N-cadherin according to the Destruction Model. Green arrows in the inset <b>A</b>′ illustrate that N-cadherin molecules are continually and rapidly synthesized and destroyed (<b>Ø</b>). Blue arrows illustrate that unpaired N-cadherin traffics via endosomes (blue) to and from the cell surface. This results in a uniform coverage of active interfaces with an equilibrium distribution of unpaired and paired molecules. (<b>B</b>) Distribution and abundance of N-cadherin according to the Recycling Model. The rates of synthesis and destruction of N-cadherin are minor relative to trafficking of N-cadherin to and from the surface (<b>B</b>′). Transport of unpaired N-cadherin through endosomes allows continual redistribution of N-cadherin along the cell surface, so that unpaired molecules have multiple opportunities to find partners to pair with. The C2 cells, with the longest active edges, exhaust their unpaired N-cadherin supply. Unpaired molecules remain in the C1 cells.</p

Timing and level of N-cadherin expression affects morphology.

<p>(<b>A</b>) Order of events (expression) in the misexpression experiment. (<b>B</b>) Simulation using the Recycling Model where the <i>N-cadherin</i> transgene initiates expression in both C2 cells and one P cell before the endogenous gene begins expression. The level of N-cadherin expression is 90% greater in the C2 cells than C1 cells (<i>N₊</i>/<i>N₀</i> = 0.9). (<b>C</b>) Simulation using the Recycling Model as in A but where the transgene and endogenous gene begin expression at the same time, and where the level of N-cadherin expression is 60% greater in C2 cells than C1 cells (<i>N₊</i>/<i>N₀</i> = 0.6). (<b>D</b>) Experimental image of ommatidium with the C2 cells and one P cell misexpressing the <i>N-cadherin</i> transgene (marked by purple).</p

Timing of N-cadherin expression with cell-cell remodeling is important for morphology.

<p>(<b>A</b>) Order of events (expression and morphogenetic) in the P cell misexpression experiment. (<b>B</b>) Simulation in which <i>N-cadherin</i> expression in P cells begins before they contact one another. All four cone cells only produce N-cadherin from its endogenous gene. Here N<i>₊</i>/N<i>₀</i> = 1.2, close to the minimum of the total error function. (<b>C</b>) Simulation in which <i>N-cadherin</i> expression in P cells begins at the same time as they contact one another. For N<i>₊</i>/N<i>₀</i> = 1.0,, the shape compares poorly with observed. This remains true for the entire range of N<i>₊</i>/N<i>₀</i> ratios. (<b>D</b>) Experimental image of ommatidium with both P cells misexpressing the <i>N-cadherin</i> transgene (marked purple). In all figures, the orange circle marks the characteristic acute angle of the mutant P/C2/P junction.</p

Nomenclature and geometry of the modeled ommatidia.

<p>Indicated are the cell types (P,C1,C2), the ommatidial scale <i>D</i>, and some of the quantities contributing to the quantification of errors: <i>L_cen</i> and <i>L_PP</i> are two examples of edge length quantities, in the case of an asymmetrically deformed ommatidium (black dashed lines) the asymmetry is quantified using <i>D</i>_x, while errors in angles between edges are expressed in terms of tension ratios. The red circle enlarges one example of a triple junction where the angles <i>θ_j</i> are used to compute the ratios <i>ρ_j</i> of the tensions <i>τ_k</i>, <i>τ_l</i> of the edges adjacent to each angle. Note that <i>E</i> cadherin is active on all edges, while on the center edge and the C1C2 edges (orange) <i>N</i> cadherin is also active.</p

Simulation of ommatidium with one <i>N-cadherin</i> mutant cone cell.

<p>(<b>A</b>) Experimental image of ommatidium with one C1 cell (left) not expressing <i>N-cadherin</i>. N-cadherin-producing cone cells are marked in purple, while cone cells not marked purple do not produce N-cadherin. Note that primary pigment cells, whether marked or not, do not normally synthesize N-cadherin. (<b>B</b>,<b>C</b>) Simulations using the Destruction (<b>B</b>) and the Recycling (<b>C</b>) Models reflect the general asymmetry and deformation of the ommatidium.. Differences between the models are slight and manifest largely in the center edge length. The width of red active edges is a measure of N-cadherin binding strength in the models. (<b>D</b>,<b>E</b>) Distribution of N-cadherin according to the Destruction (<b>D</b>) and Recycling (<b>E</b>) Models. The Destruction Model predicts unaltered densities of N-cadherin molecules and thus unchanged binding strength on the active edges. The Recycling model predicts rearrangement of the unpaired molecules in the C2 cells, leading to increased binding strength across all remaining active edges.</p

Analysis of P cell misexpression simulations.

<p>(<b>A</b>) Total error <i>F_e</i> for different simulations as a function of N<i>₊</i>/N<i>₀</i>. The red line describes the simulation error when P cells express <i>N-cadherin</i> before cone cells, and P cells do not contact each other until after the onset of N-cadherin expression in cone cells. The purple line describes simulation error when P cells express <i>N-cadherin</i> before cone cells but P cells are always in contact with each other. The black line describes simulation error when P cells and cone cells simultaneously begin <i>N-cadherin</i> expression, and P cells always are in contact with each other. (<b>B</b>) Error contributions to the best-fit model are dominated by the same <i>f_e</i> terms as in the other misexpression simulation, except that the symmetric structure of this mutant has no <i>D</i>_x contribution to the error. The minimum is located at N<i>₊</i>/N<i>₀</i>≈1.2. (<b>C</b>) The effect of N<i>₊</i>/N<i>₀</i> on the shape of the ommatidium, with only values near the error function minimum approximating the observed cruciform mutant shape. (<b>D</b>) Binding strengths of N-cadherin on various edges of the structure as N<i>₊</i>/N<i>₀</i> varies. Note the absence of binding on PP edges for N<i>₊</i>/N<i>₀</i><1.2.</p