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14

I have read somewhere on the web (I lost the reference) that the number of units (or neurons) in a hidden layer should be a power of 2 because it helps the learning algorithm to converge faster. I would quite like to see a reference to this suggestion, in case it has been misunderstood. As far as I know, there is no such effect in normal neural networks. In ...

12

This is a really common reaction after first encountering the No Free Lunch theorems (NFLs). The one for machine learning is especially unintuitive, because it flies in the face of everything that's discussed in the ML community. That said, the theorem is true, but what it means is open to some debate. To restate the theorem for people who don't know it, ...

10

A good answer to this question depends on what you want to use the labels for. When I think about "optimization," I think about a solution space and a cost function; that is, there are many possible answers that could be returned and we can know what the cost is of any particular answer. In this view, the answer is "yes"--pattern recognition is a case ...

10

I see no reason why decaying learning rates should create the kinds of jumps in losses that you are observing. It should "slow down" how quickly you "move", which in the case of a loss that otherwise consistently shrinks really should, at worst, just lead to a plateau in your losses (rather than those jumps). The first thing I observe in your code is that ...

8

TL:DR: Hyper-heuristics are metaheuristics, suited for solving the same kind of optimization problems, but (in principle) affording a "rapid prototyping" approach for non-expert practitioners. In practice, there are issues with the prevailing approach, motivating an emerging perspective on 'whitebox' hyper-heuristics. In more detail: Metaheuristics are ...

8

There are several elementary techniques to try and move a search out of the basin of attraction of local optima. They include: Probabalistically accepting worse solutions in the hope that this will jump out of the current basin (like Metropolis-Hastings acceptance in Simulated Annealing). Maintaining a list of recently-encountered states (or attributes ...

7

You can run gradient descent without back propagation, in some cases: Simple structures such as linear or logistic regression, where the gradients can be calculated directly from the inputs and cost function value. In "black box" gradient-based learning algorithms where you don't know how (or don't want to) calculate gradient analytically, so you choose to ...

7

Evolutionary algorithms (EAs) are a family of algorithms inspired by the biological evolution that can be used to solve (constrained or not) optimization problems where the function that needs to be optimized does not necessarily need to be differentiable (or satisfy any strong constraint). In EAs, you typically only need to define an encoding of the ...

7

$\max(-y_i(w x_i), 0)$ is not partial derivable respect $w$ if $w x_i=0$. Loss functions are problematic when not derivable in some point, but even more when they are flat (constant) in some interval of the weights. Assume $y_i = 1$ and $w x_i < 0$ (that is, an error of type "false negative"). In this case, function $[y_i - \text{sign}(w x_i)]^2 ... 6$g(x) = x^2$is indeed a parabola and thus has just one optimum. However, the$\text{MSE}(\boldsymbol{x}, \boldsymbol{y}) = \sum_i (y_i - f(x_i))^2$, where$\boldsymbol{x}$are the inputs,$\boldsymbol{y}$the corresponding labels and the function$f$is the model (e.g. a neural network), is not necessarily a parabola. In general, it is only a parabola if$...

5

Well, you are definitely mixing two different things. Here are those bits: The function that deep learning approximates is basically a function that best fits the INPUT DATA points. You should not think about its differentiability or optimization aspects. We don't care what type of function it is; we just want the best fit of input data (ofcourse ...

5

I can offer two (at first sight, conflicting) perspectives on this: Firstly: If the letter string 'abc' becomes 'abd' what would "doing the same thing" to 'ijk' look like? This is just one example of a problem (so-called 'letter-string analogy problems') that is not easily framed as an optimization problem - there is a range of answers that appear ...

5

Gradient descent (GD) is an optimisation algorithm, that is, it is used to find a (local) minimum of a multi-variable and differentiable function $f$. GD is an iterative and numerical optimisation algorithm. It is iterative because it proceeds in iterations. It is numerical because it is not an algorithm which produces an exact solution, due to numerical ...

5

As @nbro has already said that Hill Climbing is a family of local search algorithms. So, when you said Hill Climbing in the question I have assumed you are talking about the standard hill climbing. The standard version of hill climb has some limitations and often gets stuck in the following scenario: Local Maxima: Hill-climbing algorithm reaching on the ...

5

Empirically, I observed that algorithms like Adam and RMSProp tended to give me a final higher performance (in my case, the accuracy) on (the validation dataset) with respect to SGD. However, I also observed that Adam and RMSProp are highly sensitive to certain values of the learning rate (and, sometimes, other hyper-parameters like the batch size) and they ...

5

If the learning rate is greater than or equal to $1$ the Robbins-Monro condition $$\sum _{{t=0}}^{{\infty }}a_{t}^{2}<\infty\label{1}\tag{1},$$ where $a_t$ is the learning rate at iteration $t$, does not hold (given that a number bigger than $1$ squared becomes a bigger number), so stochastic gradient descent is not generally guaranteed to converge to a ...

4

Usually you keep track of training loss and validation loss and apply proper regularization technique (such as L1, L2, dropout, DropConnect, etc.). The more interesting technique is to observe your validation loss with respect to the number of parameters in the network (often controlled by the number of layers/feature maps). If the validation starts ...

4

What you have could be well described as a Task Allocation problem, which is studied as part of the planning subfield of AI. Chapters 10 & 11 of Russell & Norvig provide a good overview of this area, although I think they don't talk too much about Task Allocation in particular. There are two basic approaches to this problem: centralized approaches, ...

4

Hill climbing is not an algorithm, but a family of "local search" algorithms. Specific algorithms which fall into the category of "hill climbing" algorithms are 2-opt, 3-opt, 2.5-opt, 4-opt, or, in general, any N-opt. See chapter 3 of the paper "The Traveling Salesman Problem: A Case Study in Local Optimization" (by David S. Johnson and Lyle A. McGeoch) for ...

4

The "objective function" is the function that you want to minimise or maximise in your problem. The expression "objective function" is used in several different contexts (e.g. machine learning or linear programming), but it always refers to the function to be maximised or minimised in the specific (optimisation) problem. Hence, this expression is used in ...

4

In general I agree with @nbro answer, nevertheless sticking strictly to this specific question I'd like to share some speculations: what the author of the question provides us with is the Loss Function Shape so I'll try to use the full information here to compare the 2 minima looking at the LF steepness we observe the Left LM is in a steeper region than ...

4

Even the first artificial neural network - Rosenblatt's perceptron [1] had a discontinuous activation function. That network is in introductory chapters of many textbooks about AI. For example, Michael Negnevitsky. Artificial intelligence: a guide to intelligent systems. Second Edition shows how to train such networks on pages 170-174. Error backpropagation ...

3

Has this been done? Difficult to prove a negative, but I suspect although plenty of research has been done into finding ideal learning rate values (the need for learning rate at all is an annoyance), it has not been done to the level of suggesting a global function worth approximating. The problem is that learning rate tuning, like other hyperparameter ...

3

You can probably get away with a relatively low X for two reasons: The Central Limit Theorem. This tells us that the accuracy in the estimate of an agent's fitness will improve as the square root of the number of games played. In a GA, you don't need an absolute ranking of individuals, because your selection mechanism (see "related articles" here) ...

3

The differences you have observed between the two different versions of the TRPO paper are due to different formalizations of the problem and the objective. In the first version of the paper you linked, they start out in Section 2 by defining Markov Decision Processes (MDPs) as tuples that, among other things, have a cost function \$c : \mathcal{S} \...

3

For a finite value to be 'optimal,' typically you need some benefit from more paired up with some cost for more, and eventually the lines cross because the benefit decreases and the cost increases. Most models will have a reduction in error with more training data, that asymptotically approaches the best the model can do. See this image (from here) as an ...

3

This is a very large question that could be answered in a variety of ways depending on the context. For some optimization problems operating under specific conditions you can make theoretical guarantees that your optimization will solve your problem. A specific example of this is running the gradient descent algorithm on a convex function. If the function ...

3

NEAT is a genetic algorithm (GA). A genetic algorithm maintains a population of individuals (or chromosomes) and evolves it using operations like the crossover or the mutation, so that the fittest individuals keep living and most other individuals die. The nature of the individuals depends on the problem. For example, in the case of NEAT, the individuals are ...

3

Yes it has been tried. In fact there is a whole field, dubbed Genetic Programming. There is an annual competition to obtain "Human-Competitive" algorithms, and many instances of those have been found over the years.

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