Optimization
1 like170 views
The document outlines the importance of optimization in various fields, especially in quantitative finance and machine learning, highlighting how it is integral to tasks like portfolio optimization and algorithmic trading. It discusses the different types of optimization problems, such as convex and non-convex, and introduces techniques like steepest descent and stochastic gradient descent. Additionally, it emphasizes the role of optimization in enhancing model performance and the ongoing challenges and tools available in the domain.
![Optimization puts the learning in machine learning
‘Training’ = minimizing a loss function
In classification, e.g. cross-entropy = – log[p(correct label)]
In numeric, e.g. squared error = (true value – prediction)2
To train a network to identify cats
Using many photos labeled cat or not, optimize its weights to minimize
avg cross-entropy of [network’s predicted probability, true label]
Why care? Machine Learning.](https://image.slidesharecdn.com/optimization-190206170414/85/Optimization-6-320.jpg)
![Portfolio optimization
(Anish Shah 2015) Uncertain covariance and (patent-pending)
uncertainty-penalized utility
Maximize E[utility] – ɣ stdev[utility]
Deep learning - ‘dropout’
At each training optimization step, randomly omit connections so
information isn’t overfit and gets spread throughout the network
More on regularization](https://image.slidesharecdn.com/optimization-190206170414/85/Optimization-20-320.jpg)
Optimization
- 1. Topics in Optimization 2019 Copyright QuantUniversity LLC. Presented By: Anish Shah, CFA Optimization Lead QuantUniversity
- 2. • Anish leads Optimization workshops at QuantUniversity. • Anish has worked as an investment quant at the alternative data hedge fund Cargometrics, Northfield, and ITG. • He holds an MS in Applied Math from Brown University and an MS in Operations Research from the University of California, Berkeley. Anish Shah Optimization Lead 2
- 3. 3
- 4. Once you work on data, optimization is as fundamental as linear algebra and appears everywhere, if not explicitly, then under the hood Inference typically happens by maximizing a likelihood or probability OLS regression (solved analytically) median regression GARCH fitting parameters to match the data, e.g. in a Kalman Filter Why should a quant care? Inference.
- 5. Model something, add optimization, and create a “robot”, a tool that does the task automatically and better Google Maps models roads and finds short paths Expedia models flights and finds inexpensive trips Portfolio optimization models security returns’ movement and maximizes risk- adjusted return Algorithmic trading models how stock prices move in response to orders and executes trades at low cost and risk Opportunity to create and add tremendous value Why care? Build robots.
- 6. Optimization puts the learning in machine learning ‘Training’ = minimizing a loss function In classification, e.g. cross-entropy = – log[p(correct label)] In numeric, e.g. squared error = (true value – prediction)2 To train a network to identify cats Using many photos labeled cat or not, optimize its weights to minimize avg cross-entropy of [network’s predicted probability, true label] Why care? Machine Learning.
- 7. In real-world setups, no guarantee that the optimization is solvable However, particular well-studied classes have good algorithms and assurances of finding a best solution Knowing lets you restate or approximate yours as one Even if you can’t, knowing how things can fail lets you proceed skillfully Why care? Knowledge improves outcomes.
- 8. x : variables that you can play with e.g. in portfolio optimization, one entry could be % invested in AMZN in finding the shortest route, it could be 1 if a road is used, 0 if not f(x) : ‘objective function’ - criterion you want to maximize or minimize e.g. portfolio’s forecasted risk-penalized return – transaction costs expected time from point A to B – a penalty on variability h(x) ≤ b : constraints e.g. no more than 20% of portfolio in tech stocks at every node (excluding start and finish), sum roads entering = sum exits General setup
- 9. Goal: maximize expected return with a penalty on variance and costs x : vector of security weights to be found x0 : vector of initial security weights r : forecast return Σ : forecast covariance λ : risk aversion ɣ : cost aversion A, b : define linear constraints maxx rT x – λ xT Σ x – ɣ cost(x – x0) s.t. A x ≤ b Example: (vanilla) portfolio optimization
- 10. Convex optimization - minimizing a bowl shaped function under constraints that, if met by any two points, are also met by all the points on the line between them Linear programming - linear objective, linear constraints Quadratic programming - quadratic objective, linear constraints SOCP - linear objective, quadratic constraints All other convex solved by general algorithms * Note: ‘programming’ just means optimization Structures that make life easy
- 11. Linear constraints A x ≤ b Linear objective function in linear programming min cT x Quadratic objective function in quadratic programming min ½ xT Q x + cT x Linear and quadratic programming
- 12. Assorted graph problems - shortest path, maximum flow, … Algorithm is specific to the problem Dynamic programming - a general principle used when the problem separates into contingent subproblems e.g. the shortest way from NY → LA can be broken into finding 1) the shortest way from NY → every bridge that crosses the Mississippi 2) the shortest way from every bridge that crosses the Mississippi → LA Encounter things of this flavor in reinforcement learning EM (expectation maximization) - solvable in alternating steps, very particular Structures that make life easy (2)
- 13. 13 Non-convex objective functions – many extrema Non-convex constraints – path connecting acceptable points violates constraints Integer constraints or variables e.g. in portfolio optimization, trading securities in fixed sizes (round lots of 100 shares, whole futures contracts) Solved by a combination of heuristics and brute force Structures that make life hard
- 14. Relaxation - sequentially approximate as an easier problem and solve Iterate - from wherever you are, move in a direction that meets constraints and improves the objective, perhaps leaving opportunity to jump around (Metropolis) to avoid stalling at a local extremum What do you outside the easy cases?
- 15. Imagine fitting a machine learning model, with N observations of labeled data zi : data for ith observation yi : label for ith observation x : parameters (to be fit) of the model g(x, zi) : model’s output for observation i Given some loss function f, the goal is Minimizex ∑i=1..N loss(labeli, model outputi) = ∑i=1..N f(yi, g(x, zi) ) Following the gradient
- 16. Want to minimizex ∑i=1..N f(yi, g(x, zi) ) Start with an initial value of x and step to improve the objective Why not the direction of the gradient? x ← x – a ∑i=1..N ∇f(yi, g(x, zi)) Called ‘steepest descent’ in the optimization literature Was regarded as a bad idea, slowed by differences in scale Better is inverse of curvature (2nd derivatives) x gradient Following the gradient (2)
- 17. Want to minimizex ∑i=1..N f(yi, g(x, zi) ) refine x using one or a few sampled points rather than all the data x ← x – a ∇f(yk, g(x, zk)) k : the random sample a : the learning rate which is decayed over time Much lower computational cost Works as well, except for final refinement Steepest descent reemerges as stochastic gradient descent (SGD) in deep learning
- 18. Stops at ‘local maximum’ if the function maximized is not a mountain peak but a mountain range Proceeds slowly and costly (AWS $) because of a poorly tuned or inappropriate algorithm Chases noise or overfits yielding poor out-of-sample results How can optimization go wrong?
- 19. Suppose the goal is minx f(x, y) where y is noisy data Robust optimization - finds the best worst case given a range of inputs minx maxy f(x, y) where y, rather than being fixed, takes value in a range hard to solve except for special cases Regularization - penalizes the norm of the decision variable minx f(x, y) + λ |x| Ridge regression, lasso penalty, ... Tightly connected to Bayesian statistics Addressing noise and overfitting
- 20. Portfolio optimization (Anish Shah 2015) Uncertain covariance and (patent-pending) uncertainty-penalized utility Maximize E[utility] – ɣ stdev[utility] Deep learning - ‘dropout’ At each training optimization step, randomly omit connections so information isn’t overfit and gets spread throughout the network More on regularization
- 21. 21 • cvxopt – convex optimizer library • nlopt – nonlinear optimizer library • CBC – mixed integer programming library • Google OR Tools – lp and assorted graph algorithms • CPLEX – commercial optimizer for the hardest problems Optimization tools
- 22. 22 • Portfolio optimization • Algorithmic trading • Index/ETF construction • Hedging • Asset/liability matching • Nowcasting covariance Applications in Finance
- 23. Optimization is everywhere, especially as ML becomes pervasive This is the direction quant is headed Knowledge lets you create and add more value The optimization piece of ML doesn’t seem fully matured – there are puzzles to be solved! Summary